| L(s) = 1 | + 2·3-s + 5-s − 2·7-s + 9-s − 4·11-s + 2·15-s + 2·17-s − 4·21-s − 6·23-s + 25-s − 4·27-s − 10·29-s − 8·33-s − 2·35-s − 10·37-s + 2·41-s + 2·43-s + 45-s + 6·47-s − 3·49-s + 4·51-s + 2·53-s − 4·55-s + 8·59-s + 2·61-s − 2·63-s + 6·67-s + ⋯ |
| L(s) = 1 | + 1.15·3-s + 0.447·5-s − 0.755·7-s + 1/3·9-s − 1.20·11-s + 0.516·15-s + 0.485·17-s − 0.872·21-s − 1.25·23-s + 1/5·25-s − 0.769·27-s − 1.85·29-s − 1.39·33-s − 0.338·35-s − 1.64·37-s + 0.312·41-s + 0.304·43-s + 0.149·45-s + 0.875·47-s − 3/7·49-s + 0.560·51-s + 0.274·53-s − 0.539·55-s + 1.04·59-s + 0.256·61-s − 0.251·63-s + 0.733·67-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3380 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3380 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 5 | \( 1 - T \) |
| 13 | \( 1 \) |
| good | 3 | \( 1 - 2 T + p T^{2} \) |
| 7 | \( 1 + 2 T + p T^{2} \) |
| 11 | \( 1 + 4 T + p T^{2} \) |
| 17 | \( 1 - 2 T + p T^{2} \) |
| 19 | \( 1 + p T^{2} \) |
| 23 | \( 1 + 6 T + p T^{2} \) |
| 29 | \( 1 + 10 T + p T^{2} \) |
| 31 | \( 1 + p T^{2} \) |
| 37 | \( 1 + 10 T + p T^{2} \) |
| 41 | \( 1 - 2 T + p T^{2} \) |
| 43 | \( 1 - 2 T + p T^{2} \) |
| 47 | \( 1 - 6 T + p T^{2} \) |
| 53 | \( 1 - 2 T + p T^{2} \) |
| 59 | \( 1 - 8 T + p T^{2} \) |
| 61 | \( 1 - 2 T + p T^{2} \) |
| 67 | \( 1 - 6 T + p T^{2} \) |
| 71 | \( 1 - 8 T + p T^{2} \) |
| 73 | \( 1 + 10 T + p T^{2} \) |
| 79 | \( 1 + 16 T + p T^{2} \) |
| 83 | \( 1 + 6 T + p T^{2} \) |
| 89 | \( 1 + 10 T + p T^{2} \) |
| 97 | \( 1 + 2 T + p T^{2} \) |
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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
−8.294611322662235322161326445331, −7.60772906550111507201069656862, −6.96248036595297536352932838854, −5.77983111874602530633481360255, −5.44430771612759995461563322676, −4.06833875739798601326005889963, −3.35972004290434705093125532703, −2.57960849572452519565380124939, −1.84721977871400065455522858092, 0,
1.84721977871400065455522858092, 2.57960849572452519565380124939, 3.35972004290434705093125532703, 4.06833875739798601326005889963, 5.44430771612759995461563322676, 5.77983111874602530633481360255, 6.96248036595297536352932838854, 7.60772906550111507201069656862, 8.294611322662235322161326445331
