| L(s) = 1 | − 4-s + 4·5-s − 9-s − 8·11-s + 16-s + 2·19-s − 4·20-s + 11·25-s + 4·29-s + 36-s − 24·41-s + 8·44-s − 4·45-s + 14·49-s − 32·55-s + 20·59-s − 12·61-s − 64-s + 24·71-s − 2·76-s + 16·79-s + 4·80-s + 81-s + 16·89-s + 8·95-s + 8·99-s − 11·100-s + ⋯ |
| L(s) = 1 | − 1/2·4-s + 1.78·5-s − 1/3·9-s − 2.41·11-s + 1/4·16-s + 0.458·19-s − 0.894·20-s + 11/5·25-s + 0.742·29-s + 1/6·36-s − 3.74·41-s + 1.20·44-s − 0.596·45-s + 2·49-s − 4.31·55-s + 2.60·59-s − 1.53·61-s − 1/8·64-s + 2.84·71-s − 0.229·76-s + 1.80·79-s + 0.447·80-s + 1/9·81-s + 1.69·89-s + 0.820·95-s + 0.804·99-s − 1.09·100-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 324900 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 324900 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.754878071\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.754878071\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_2$ | \( 1 + T^{2} \) |
| 3 | $C_2$ | \( 1 + T^{2} \) |
| 5 | $C_2$ | \( 1 - 4 T + p T^{2} \) |
| 19 | $C_1$ | \( ( 1 - T )^{2} \) |
| good | 7 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 11 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 13 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 17 | $C_2^2$ | \( 1 + 2 T^{2} + p^{2} T^{4} \) |
| 23 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 29 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 37 | $C_2^2$ | \( 1 - 58 T^{2} + p^{2} T^{4} \) |
| 41 | $C_2$ | \( ( 1 + 12 T + p T^{2} )^{2} \) |
| 43 | $C_2^2$ | \( 1 - 50 T^{2} + p^{2} T^{4} \) |
| 47 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 53 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 59 | $C_2$ | \( ( 1 - 10 T + p T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 67 | $C_2^2$ | \( 1 - 118 T^{2} + p^{2} T^{4} \) |
| 71 | $C_2$ | \( ( 1 - 12 T + p T^{2} )^{2} \) |
| 73 | $C_2^2$ | \( 1 - 82 T^{2} + p^{2} T^{4} \) |
| 79 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| 83 | $C_2^2$ | \( 1 - 22 T^{2} + p^{2} T^{4} \) |
| 89 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| 97 | $C_2^2$ | \( 1 - 94 T^{2} + p^{2} T^{4} \) |
| show more | | |
| show less | | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−10.62377838197558305350434494111, −10.34091002365277322260252363673, −10.23329464756720269519501952407, −9.760495732942717985819936591379, −9.288803807794320598797227422516, −8.609356843080483474480513949284, −8.565529098615976236460109950961, −7.899980267027830944401240189752, −7.47683564774457667421793465899, −6.73519482891855758349620952875, −6.46297204943467024401033728284, −5.68045820068518689822331327989, −5.40969598714047347836151956300, −5.01680085486630376879797109998, −4.78021947125316864590275242887, −3.55211255126426616751106338385, −3.12160359901020048538613695460, −2.26945513601220936872196404569, −2.10180563193725297307983955965, −0.74937687554549699097085119247,
0.74937687554549699097085119247, 2.10180563193725297307983955965, 2.26945513601220936872196404569, 3.12160359901020048538613695460, 3.55211255126426616751106338385, 4.78021947125316864590275242887, 5.01680085486630376879797109998, 5.40969598714047347836151956300, 5.68045820068518689822331327989, 6.46297204943467024401033728284, 6.73519482891855758349620952875, 7.47683564774457667421793465899, 7.899980267027830944401240189752, 8.565529098615976236460109950961, 8.609356843080483474480513949284, 9.288803807794320598797227422516, 9.760495732942717985819936591379, 10.23329464756720269519501952407, 10.34091002365277322260252363673, 10.62377838197558305350434494111
