| L(s) = 1 | + 74·19-s − 92·31-s − 23·49-s + 94·61-s − 22·79-s + 428·109-s + 242·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s − 191·169-s + 173-s + 179-s + 181-s + 191-s + 193-s + 197-s + 199-s + 211-s + 223-s + 227-s + ⋯ |
| L(s) = 1 | + 3.89·19-s − 2.96·31-s − 0.469·49-s + 1.54·61-s − 0.278·79-s + 3.92·109-s + 2·121-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 − 1.13·169-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 + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 7290000 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7290000 ^{s/2} \, \Gamma_{\C}(s+1)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(3.790869849\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.790869849\) |
| \(L(2)\) |
|
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 | | \( 1 \) |
| 5 | | \( 1 \) |
| good | 7 | $C_2^2$ | \( 1 + 23 T^{2} + p^{4} T^{4} \) |
| 11 | $C_1$$\times$$C_1$ | \( ( 1 - p T )^{2}( 1 + p T )^{2} \) |
| 13 | $C_2^2$ | \( 1 + 191 T^{2} + p^{4} T^{4} \) |
| 17 | $C_2$ | \( ( 1 + p^{2} T^{2} )^{2} \) |
| 19 | $C_2$ | \( ( 1 - 37 T + p^{2} T^{2} )^{2} \) |
| 23 | $C_2$ | \( ( 1 + p^{2} T^{2} )^{2} \) |
| 29 | $C_1$$\times$$C_1$ | \( ( 1 - p T )^{2}( 1 + p T )^{2} \) |
| 31 | $C_2$ | \( ( 1 + 46 T + p^{2} T^{2} )^{2} \) |
| 37 | $C_2^2$ | \( 1 + 2591 T^{2} + p^{4} T^{4} \) |
| 41 | $C_1$$\times$$C_1$ | \( ( 1 - p T )^{2}( 1 + p T )^{2} \) |
| 43 | $C_2^2$ | \( 1 - 3214 T^{2} + p^{4} T^{4} \) |
| 47 | $C_2$ | \( ( 1 + p^{2} T^{2} )^{2} \) |
| 53 | $C_2$ | \( ( 1 + p^{2} T^{2} )^{2} \) |
| 59 | $C_1$$\times$$C_1$ | \( ( 1 - p T )^{2}( 1 + p T )^{2} \) |
| 61 | $C_2$ | \( ( 1 - 47 T + p^{2} T^{2} )^{2} \) |
| 67 | $C_2^2$ | \( 1 - 8809 T^{2} + p^{4} T^{4} \) |
| 71 | $C_1$$\times$$C_1$ | \( ( 1 - p T )^{2}( 1 + p T )^{2} \) |
| 73 | $C_2^2$ | \( 1 + 9791 T^{2} + p^{4} T^{4} \) |
| 79 | $C_2$ | \( ( 1 + 11 T + p^{2} T^{2} )^{2} \) |
| 83 | $C_2$ | \( ( 1 + p^{2} T^{2} )^{2} \) |
| 89 | $C_1$$\times$$C_1$ | \( ( 1 - p T )^{2}( 1 + p T )^{2} \) |
| 97 | $C_2^2$ | \( 1 + 9743 T^{2} + p^{4} T^{4} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−8.758968719040413724159401856961, −8.693029164915622428680294124128, −7.80089769820038421778627656763, −7.79418755598108967235016244840, −7.31492429688172838923953167701, −7.05555223400700319680050632608, −6.79106316613139715418890348559, −5.96063484352738532636700755573, −5.57370813926779999821188003884, −5.56486949655109923383470123456, −5.00438938029269444775855464466, −4.66288401847522957918206969994, −3.95636778317204524316573817965, −3.51695428250784421167031966582, −3.20782584247536254999097402133, −2.92506605067315971924054493644, −1.93872028989534773688614048140, −1.76915795043288088701492266856, −0.913405412043670330867599824668, −0.55998066346963596578799167088,
0.55998066346963596578799167088, 0.913405412043670330867599824668, 1.76915795043288088701492266856, 1.93872028989534773688614048140, 2.92506605067315971924054493644, 3.20782584247536254999097402133, 3.51695428250784421167031966582, 3.95636778317204524316573817965, 4.66288401847522957918206969994, 5.00438938029269444775855464466, 5.56486949655109923383470123456, 5.57370813926779999821188003884, 5.96063484352738532636700755573, 6.79106316613139715418890348559, 7.05555223400700319680050632608, 7.31492429688172838923953167701, 7.79418755598108967235016244840, 7.80089769820038421778627656763, 8.693029164915622428680294124128, 8.758968719040413724159401856961
