| L(s) = 1 | + 4·5-s − 8·11-s + 3·13-s − 7·17-s + 5·19-s − 8·23-s + 2·25-s − 29-s − 3·31-s − 11·37-s + 9·41-s − 5·43-s − 3·47-s + 3·53-s − 32·55-s + 7·59-s + 3·61-s + 12·65-s − 13·67-s + 16·71-s + 7·73-s + 9·79-s − 83-s − 28·85-s − 15·89-s + 20·95-s − 17·97-s + ⋯ |
| L(s) = 1 | + 1.78·5-s − 2.41·11-s + 0.832·13-s − 1.69·17-s + 1.14·19-s − 1.66·23-s + 2/5·25-s − 0.185·29-s − 0.538·31-s − 1.80·37-s + 1.40·41-s − 0.762·43-s − 0.437·47-s + 0.412·53-s − 4.31·55-s + 0.911·59-s + 0.384·61-s + 1.48·65-s − 1.58·67-s + 1.89·71-s + 0.819·73-s + 1.01·79-s − 0.109·83-s − 3.03·85-s − 1.58·89-s + 2.05·95-s − 1.72·97-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 28005264 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 28005264 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.5618918768\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.5618918768\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| good | 5 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 11 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 13 | $C_2^2$ | \( 1 - 3 T - 4 T^{2} - 3 p T^{3} + p^{2} T^{4} \) |
| 17 | $C_2^2$ | \( 1 + 7 T + 32 T^{2} + 7 p T^{3} + p^{2} T^{4} \) |
| 19 | $C_2^2$ | \( 1 - 5 T + 6 T^{2} - 5 p T^{3} + p^{2} T^{4} \) |
| 23 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 29 | $C_2^2$ | \( 1 + T - 28 T^{2} + p T^{3} + p^{2} T^{4} \) |
| 31 | $C_2^2$ | \( 1 + 3 T - 22 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 37 | $C_2$ | \( ( 1 + T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 41 | $C_2^2$ | \( 1 - 9 T + 40 T^{2} - 9 p T^{3} + p^{2} T^{4} \) |
| 43 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 13 T + p T^{2} ) \) |
| 47 | $C_2^2$ | \( 1 + 3 T - 38 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 53 | $C_2^2$ | \( 1 - 3 T - 44 T^{2} - 3 p T^{3} + p^{2} T^{4} \) |
| 59 | $C_2^2$ | \( 1 - 7 T - 10 T^{2} - 7 p T^{3} + p^{2} T^{4} \) |
| 61 | $C_2^2$ | \( 1 - 3 T - 52 T^{2} - 3 p T^{3} + p^{2} T^{4} \) |
| 67 | $C_2^2$ | \( 1 + 13 T + 102 T^{2} + 13 p T^{3} + p^{2} T^{4} \) |
| 71 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 - 17 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 79 | $C_2^2$ | \( 1 - 9 T + 2 T^{2} - 9 p T^{3} + p^{2} T^{4} \) |
| 83 | $C_2^2$ | \( 1 + T - 82 T^{2} + p T^{3} + p^{2} T^{4} \) |
| 89 | $C_2^2$ | \( 1 + 15 T + 136 T^{2} + 15 p T^{3} + p^{2} T^{4} \) |
| 97 | $C_2^2$ | \( 1 + 17 T + 192 T^{2} + 17 p T^{3} + p^{2} T^{4} \) |
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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
−8.471569299830570261885680814790, −7.951161270402134281175056581958, −7.76256697761775094281284637422, −7.34737295339234893894023166121, −6.83801501672938365496937015119, −6.48005740464511607959397181931, −6.22261409301363136109878713499, −5.76244415660415257037642647171, −5.32699039620820098158153685314, −5.30912497191537879810876967483, −5.07883051308105902858202243707, −4.10773982465257530835407954156, −4.09738265007982544380199274375, −3.45223248475053897592685082813, −2.86701839339111701781496624199, −2.39540481917874066439325480381, −2.24866142287559010084300251927, −1.77699553025560590127593494969, −1.28688065587659384193810300401, −0.18111829181898944349505268921,
0.18111829181898944349505268921, 1.28688065587659384193810300401, 1.77699553025560590127593494969, 2.24866142287559010084300251927, 2.39540481917874066439325480381, 2.86701839339111701781496624199, 3.45223248475053897592685082813, 4.09738265007982544380199274375, 4.10773982465257530835407954156, 5.07883051308105902858202243707, 5.30912497191537879810876967483, 5.32699039620820098158153685314, 5.76244415660415257037642647171, 6.22261409301363136109878713499, 6.48005740464511607959397181931, 6.83801501672938365496937015119, 7.34737295339234893894023166121, 7.76256697761775094281284637422, 7.951161270402134281175056581958, 8.471569299830570261885680814790
