| L(s) = 1 | − 3-s + 9-s + 2·11-s − 13-s + 2·17-s + 5·19-s − 6·23-s − 5·25-s − 27-s − 8·29-s − 3·31-s − 2·33-s − 9·37-s + 39-s − 2·41-s − 43-s + 8·47-s − 2·51-s + 6·53-s − 5·57-s + 6·59-s + 2·61-s + 5·67-s + 6·69-s − 4·71-s + 11·73-s + 5·75-s + ⋯ |
| L(s) = 1 | − 0.577·3-s + 1/3·9-s + 0.603·11-s − 0.277·13-s + 0.485·17-s + 1.14·19-s − 1.25·23-s − 25-s − 0.192·27-s − 1.48·29-s − 0.538·31-s − 0.348·33-s − 1.47·37-s + 0.160·39-s − 0.312·41-s − 0.152·43-s + 1.16·47-s − 0.280·51-s + 0.824·53-s − 0.662·57-s + 0.781·59-s + 0.256·61-s + 0.610·67-s + 0.722·69-s − 0.474·71-s + 1.28·73-s + 0.577·75-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 | \( 1 + p T^{2} \) |
| 11 | \( 1 - 2 T + p T^{2} \) |
| 13 | \( 1 + T + p T^{2} \) |
| 17 | \( 1 - 2 T + p T^{2} \) |
| 19 | \( 1 - 5 T + p T^{2} \) |
| 23 | \( 1 + 6 T + p T^{2} \) |
| 29 | \( 1 + 8 T + p T^{2} \) |
| 31 | \( 1 + 3 T + p T^{2} \) |
| 37 | \( 1 + 9 T + p T^{2} \) |
| 41 | \( 1 + 2 T + p T^{2} \) |
| 43 | \( 1 + T + p T^{2} \) |
| 47 | \( 1 - 8 T + p T^{2} \) |
| 53 | \( 1 - 6 T + p T^{2} \) |
| 59 | \( 1 - 6 T + p T^{2} \) |
| 61 | \( 1 - 2 T + p T^{2} \) |
| 67 | \( 1 - 5 T + p T^{2} \) |
| 71 | \( 1 + 4 T + p T^{2} \) |
| 73 | \( 1 - 11 T + p T^{2} \) |
| 79 | \( 1 - 5 T + p T^{2} \) |
| 83 | \( 1 + p T^{2} \) |
| 89 | \( 1 + 12 T + p T^{2} \) |
| 97 | \( 1 + 18 T + p T^{2} \) |
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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
−7.75179850145786973670845167576, −7.26041845583295749804986654665, −6.47369845569624903032123188782, −5.53455818841724045449973117314, −5.31870148242366789484744602065, −3.98942434578535227021858120933, −3.63536019108386244487276468779, −2.26110296965952128561841507252, −1.34136399448656466355096540218, 0,
1.34136399448656466355096540218, 2.26110296965952128561841507252, 3.63536019108386244487276468779, 3.98942434578535227021858120933, 5.31870148242366789484744602065, 5.53455818841724045449973117314, 6.47369845569624903032123188782, 7.26041845583295749804986654665, 7.75179850145786973670845167576
