| L(s) = 1 | − 3-s − 2·5-s + 9-s − 2·13-s + 2·15-s − 2·17-s + 4·19-s − 25-s − 27-s + 6·29-s + 6·37-s + 2·39-s + 6·41-s − 8·43-s − 2·45-s + 8·47-s + 2·51-s + 6·53-s − 4·57-s − 12·59-s − 10·61-s + 4·65-s − 16·67-s + 8·71-s + 6·73-s + 75-s − 8·79-s + ⋯ |
| L(s) = 1 | − 0.577·3-s − 0.894·5-s + 1/3·9-s − 0.554·13-s + 0.516·15-s − 0.485·17-s + 0.917·19-s − 1/5·25-s − 0.192·27-s + 1.11·29-s + 0.986·37-s + 0.320·39-s + 0.937·41-s − 1.21·43-s − 0.298·45-s + 1.16·47-s + 0.280·51-s + 0.824·53-s − 0.529·57-s − 1.56·59-s − 1.28·61-s + 0.496·65-s − 1.95·67-s + 0.949·71-s + 0.702·73-s + 0.115·75-s − 0.900·79-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4704 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4704 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + T \) |
| 7 | \( 1 \) |
| good | 5 | \( 1 + 2 T + p T^{2} \) |
| 11 | \( 1 + p T^{2} \) |
| 13 | \( 1 + 2 T + p T^{2} \) |
| 17 | \( 1 + 2 T + p T^{2} \) |
| 19 | \( 1 - 4 T + p T^{2} \) |
| 23 | \( 1 + p T^{2} \) |
| 29 | \( 1 - 6 T + p T^{2} \) |
| 31 | \( 1 + p T^{2} \) |
| 37 | \( 1 - 6 T + p T^{2} \) |
| 41 | \( 1 - 6 T + p T^{2} \) |
| 43 | \( 1 + 8 T + p T^{2} \) |
| 47 | \( 1 - 8 T + p T^{2} \) |
| 53 | \( 1 - 6 T + p T^{2} \) |
| 59 | \( 1 + 12 T + p T^{2} \) |
| 61 | \( 1 + 10 T + p T^{2} \) |
| 67 | \( 1 + 16 T + p T^{2} \) |
| 71 | \( 1 - 8 T + p T^{2} \) |
| 73 | \( 1 - 6 T + p T^{2} \) |
| 79 | \( 1 + 8 T + p T^{2} \) |
| 83 | \( 1 + 12 T + p T^{2} \) |
| 89 | \( 1 - 14 T + p T^{2} \) |
| 97 | \( 1 - 6 T + p T^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−7.66142913375025115208178290364, −7.42597423638088706386566068840, −6.46128413861177370097029168596, −5.79654630775654302973327961062, −4.82010862279680948225099374472, −4.35754691737531965446674586814, −3.40492145020932294397956269601, −2.48579877676909362685142874621, −1.14396888799931864879857010749, 0,
1.14396888799931864879857010749, 2.48579877676909362685142874621, 3.40492145020932294397956269601, 4.35754691737531965446674586814, 4.82010862279680948225099374472, 5.79654630775654302973327961062, 6.46128413861177370097029168596, 7.42597423638088706386566068840, 7.66142913375025115208178290364
