L(s) = 1 | − 100·25-s − 184·41-s + 196·49-s − 34·81-s − 392·113-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s − 676·169-s + 173-s + 179-s + 181-s + 191-s + 193-s + 197-s + 199-s + 211-s + 223-s + 227-s + 229-s + 233-s + ⋯ |
L(s) = 1 | − 4·25-s − 4.48·41-s + 4·49-s − 0.419·81-s − 3.46·113-s + 0.00787·127-s + 0.00763·131-s + 0.00729·137-s + 0.00719·139-s + 0.00671·149-s + 0.00662·151-s + 0.00636·157-s + 0.00613·163-s + 0.00598·167-s − 4·169-s + 0.00578·173-s + 0.00558·179-s + 0.00552·181-s + 0.00523·191-s + 0.00518·193-s + 0.00507·197-s + 0.00502·199-s + 0.00473·211-s + 0.00448·223-s + 0.00440·227-s + 0.00436·229-s + 0.00429·233-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{36}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(3-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{36}\right)^{s/2} \, \Gamma_{\C}(s+1)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
\(L(\frac{3}{2})\) |
\(\approx\) |
\(0.1618257165\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.1618257165\) |
\(L(2)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
---|
bad | 2 | | \( 1 \) |
good | 3 | $C_2^2$$\times$$C_2^2$ | \( ( 1 - 8 T + 32 T^{2} - 8 p^{2} T^{3} + p^{4} T^{4} )( 1 + 8 T + 32 T^{2} + 8 p^{2} T^{3} + p^{4} T^{4} ) \) |
| 5 | $C_2$ | \( ( 1 + p^{2} T^{2} )^{4} \) |
| 7 | $C_1$$\times$$C_1$ | \( ( 1 - p T )^{4}( 1 + p T )^{4} \) |
| 11 | $C_2^2$$\times$$C_2^2$ | \( ( 1 - 24 T + 288 T^{2} - 24 p^{2} T^{3} + p^{4} T^{4} )( 1 + 24 T + 288 T^{2} + 24 p^{2} T^{3} + p^{4} T^{4} ) \) |
| 13 | $C_2$ | \( ( 1 + p^{2} T^{2} )^{4} \) |
| 17 | $C_2^2$ | \( ( 1 - 574 T^{2} + p^{4} T^{4} )^{2} \) |
| 19 | $C_2^2$$\times$$C_2^2$ | \( ( 1 - 24 T + 288 T^{2} - 24 p^{2} T^{3} + p^{4} T^{4} )( 1 + 24 T + 288 T^{2} + 24 p^{2} T^{3} + p^{4} T^{4} ) \) |
| 23 | $C_1$$\times$$C_1$ | \( ( 1 - p T )^{4}( 1 + p T )^{4} \) |
| 29 | $C_2$ | \( ( 1 + p^{2} T^{2} )^{4} \) |
| 31 | $C_1$$\times$$C_1$ | \( ( 1 - p T )^{4}( 1 + p T )^{4} \) |
| 37 | $C_2$ | \( ( 1 + p^{2} T^{2} )^{4} \) |
| 41 | $C_2$ | \( ( 1 + 46 T + p^{2} T^{2} )^{4} \) |
| 43 | $C_2^2$$\times$$C_2^2$ | \( ( 1 - 120 T + 7200 T^{2} - 120 p^{2} T^{3} + p^{4} T^{4} )( 1 + 120 T + 7200 T^{2} + 120 p^{2} T^{3} + p^{4} T^{4} ) \) |
| 47 | $C_1$$\times$$C_1$ | \( ( 1 - p T )^{4}( 1 + p T )^{4} \) |
| 53 | $C_2$ | \( ( 1 + p^{2} T^{2} )^{4} \) |
| 59 | $C_2^2$$\times$$C_2^2$ | \( ( 1 - 120 T + 7200 T^{2} - 120 p^{2} T^{3} + p^{4} T^{4} )( 1 + 120 T + 7200 T^{2} + 120 p^{2} T^{3} + p^{4} T^{4} ) \) |
| 61 | $C_2$ | \( ( 1 + p^{2} T^{2} )^{4} \) |
| 67 | $C_2^2$$\times$$C_2^2$ | \( ( 1 - 168 T + 14112 T^{2} - 168 p^{2} T^{3} + p^{4} T^{4} )( 1 + 168 T + 14112 T^{2} + 168 p^{2} T^{3} + p^{4} T^{4} ) \) |
| 71 | $C_1$$\times$$C_1$ | \( ( 1 - p T )^{4}( 1 + p T )^{4} \) |
| 73 | $C_2^2$ | \( ( 1 + 9506 T^{2} + p^{4} T^{4} )^{2} \) |
| 79 | $C_1$$\times$$C_1$ | \( ( 1 - p T )^{4}( 1 + p T )^{4} \) |
| 83 | $C_2^2$$\times$$C_2^2$ | \( ( 1 - 72 T + 2592 T^{2} - 72 p^{2} T^{3} + p^{4} T^{4} )( 1 + 72 T + 2592 T^{2} + 72 p^{2} T^{3} + p^{4} T^{4} ) \) |
| 89 | $C_2^2$ | \( ( 1 + 5474 T^{2} + p^{4} T^{4} )^{2} \) |
| 97 | $C_2^2$ | \( ( 1 - 9982 T^{2} + p^{4} T^{4} )^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{8} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−7.54496201369519345856615396567, −7.49362089085268365121486626597, −7.06159412350939683916441950433, −7.02284354612260863186035728462, −6.75285754624906009430192148529, −6.48048178937702194810551889683, −6.05266609481204406546807013447, −5.90663477350550173092647195708, −5.61085326408028744582661215445, −5.53204424800892812733793805725, −5.29059240128494167249971274028, −4.84614562490900313773014768993, −4.61039881817635448747192758121, −4.20976214986503029283208650376, −4.04567183131378433833147501824, −3.73589701087092884285629259680, −3.47291560065710072051790045442, −3.30970439351563337479392992959, −2.74736990690413497296960595414, −2.34024367121887577327266669878, −2.11033319827092539985263490734, −1.74850958804639383578086825162, −1.45923902308736539313932287318, −0.78092502109633740070459906789, −0.083407147722585489702499989776,
0.083407147722585489702499989776, 0.78092502109633740070459906789, 1.45923902308736539313932287318, 1.74850958804639383578086825162, 2.11033319827092539985263490734, 2.34024367121887577327266669878, 2.74736990690413497296960595414, 3.30970439351563337479392992959, 3.47291560065710072051790045442, 3.73589701087092884285629259680, 4.04567183131378433833147501824, 4.20976214986503029283208650376, 4.61039881817635448747192758121, 4.84614562490900313773014768993, 5.29059240128494167249971274028, 5.53204424800892812733793805725, 5.61085326408028744582661215445, 5.90663477350550173092647195708, 6.05266609481204406546807013447, 6.48048178937702194810551889683, 6.75285754624906009430192148529, 7.02284354612260863186035728462, 7.06159412350939683916441950433, 7.49362089085268365121486626597, 7.54496201369519345856615396567