| L(s) = 1 | − 40·13-s + 100·25-s + 112·37-s + 14·49-s − 200·61-s − 456·73-s + 184·97-s + 256·109-s + 456·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 324·169-s + 173-s + 179-s + 181-s + 191-s + 193-s + 197-s + 199-s + 211-s + ⋯ |
| L(s) = 1 | − 3.07·13-s + 4·25-s + 3.02·37-s + 2/7·49-s − 3.27·61-s − 6.24·73-s + 1.89·97-s + 2.34·109-s + 3.76·121-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 + 1.91·169-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 + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{20} \cdot 3^{8} \cdot 7^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(3-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{20} \cdot 3^{8} \cdot 7^{4}\right)^{s/2} \, \Gamma_{\C}(s+1)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(1.569850406\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.569850406\) |
| \(L(2)\) |
|
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 | | \( 1 \) |
| 3 | | \( 1 \) |
| 7 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| good | 5 | $C_1$$\times$$C_1$ | \( ( 1 - p T )^{4}( 1 + p T )^{4} \) |
| 11 | $C_2^2$ | \( ( 1 - 228 T^{2} + p^{4} T^{4} )^{2} \) |
| 13 | $C_2$ | \( ( 1 + 10 T + p^{2} T^{2} )^{4} \) |
| 17 | $C_2^2$ | \( ( 1 - 186 T^{2} + p^{4} T^{4} )^{2} \) |
| 19 | $C_2^2$ | \( ( 1 - 286 T^{2} + p^{4} T^{4} )^{2} \) |
| 23 | $C_2^2$ | \( ( 1 - 1044 T^{2} + p^{4} T^{4} )^{2} \) |
| 29 | $C_2^2$ | \( ( 1 + p^{4} T^{4} )^{2} \) |
| 31 | $C_2^2$ | \( ( 1 + 130 T^{2} + p^{4} T^{4} )^{2} \) |
| 37 | $C_2$ | \( ( 1 - 28 T + p^{2} T^{2} )^{4} \) |
| 41 | $C_2^2$ | \( ( 1 - 1314 T^{2} + p^{4} T^{4} )^{2} \) |
| 43 | $C_2^2$ | \( ( 1 - 3470 T^{2} + p^{4} T^{4} )^{2} \) |
| 47 | $C_2^2$ | \( ( 1 - 3018 T^{2} + p^{4} T^{4} )^{2} \) |
| 53 | $C_2^2$ | \( ( 1 + 1344 T^{2} + p^{4} T^{4} )^{2} \) |
| 59 | $C_2^2$ | \( ( 1 + 2502 T^{2} + p^{4} T^{4} )^{2} \) |
| 61 | $C_2$ | \( ( 1 + 50 T + p^{2} T^{2} )^{4} \) |
| 67 | $C_2^2$ | \( ( 1 + 7606 T^{2} + p^{4} T^{4} )^{2} \) |
| 71 | $C_2^2$ | \( ( 1 + 124 T^{2} + p^{4} T^{4} )^{2} \) |
| 73 | $C_2$ | \( ( 1 + 114 T + p^{2} T^{2} )^{4} \) |
| 79 | $C_2^2$ | \( ( 1 + 12454 T^{2} + p^{4} T^{4} )^{2} \) |
| 83 | $C_2^2$ | \( ( 1 - 13554 T^{2} + p^{4} T^{4} )^{2} \) |
| 89 | $C_2^2$ | \( ( 1 - 13250 T^{2} + p^{4} T^{4} )^{2} \) |
| 97 | $C_2$ | \( ( 1 - 46 T + p^{2} T^{2} )^{4} \) |
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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
−6.33855858696545396097253433720, −6.03662696897819909329504696146, −5.92175161228006082547509822643, −5.86182841325231833728686845168, −5.35890148288060029794478835583, −5.20404614194693041518612660200, −4.87766946059034745275281582473, −4.79895261639775032583830757121, −4.60029122322915385857833205127, −4.41864002147764700268173092761, −4.37573139899742121034736054121, −4.20257280505552479688823625855, −3.45543621373733709989556834531, −3.36191013145332610194089438698, −2.99690123243621535651676583395, −2.91480551381234311820466541451, −2.83125112993786993690535569792, −2.48939873324373921311073404971, −2.21796857416815817364395056026, −1.98313755378496668426674008330, −1.45586370595338742876491578358, −1.29187424799546897563167786285, −0.869291315711907992906353206997, −0.58782868034476045880787952918, −0.17940287602207278739669593241,
0.17940287602207278739669593241, 0.58782868034476045880787952918, 0.869291315711907992906353206997, 1.29187424799546897563167786285, 1.45586370595338742876491578358, 1.98313755378496668426674008330, 2.21796857416815817364395056026, 2.48939873324373921311073404971, 2.83125112993786993690535569792, 2.91480551381234311820466541451, 2.99690123243621535651676583395, 3.36191013145332610194089438698, 3.45543621373733709989556834531, 4.20257280505552479688823625855, 4.37573139899742121034736054121, 4.41864002147764700268173092761, 4.60029122322915385857833205127, 4.79895261639775032583830757121, 4.87766946059034745275281582473, 5.20404614194693041518612660200, 5.35890148288060029794478835583, 5.86182841325231833728686845168, 5.92175161228006082547509822643, 6.03662696897819909329504696146, 6.33855858696545396097253433720
