L(s) = 1 | + 3-s + 2·5-s + 9-s − 2·13-s + 2·15-s − 6·17-s − 4·19-s − 4·23-s − 25-s + 27-s − 6·29-s + 8·31-s + 10·37-s − 2·39-s + 10·41-s − 12·43-s + 2·45-s + 8·47-s − 6·51-s − 6·53-s − 4·57-s + 4·59-s − 10·61-s − 4·65-s − 12·67-s − 4·69-s + 4·71-s + ⋯ |
L(s) = 1 | + 0.577·3-s + 0.894·5-s + 1/3·9-s − 0.554·13-s + 0.516·15-s − 1.45·17-s − 0.917·19-s − 0.834·23-s − 1/5·25-s + 0.192·27-s − 1.11·29-s + 1.43·31-s + 1.64·37-s − 0.320·39-s + 1.56·41-s − 1.82·43-s + 0.298·45-s + 1.16·47-s − 0.840·51-s − 0.824·53-s − 0.529·57-s + 0.520·59-s − 1.28·61-s − 0.496·65-s − 1.46·67-s − 0.481·69-s + 0.474·71-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 9408 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 9408 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + 6 T + p T^{2} \) |
| 19 | \( 1 + 4 T + p T^{2} \) |
| 23 | \( 1 + 4 T + p T^{2} \) |
| 29 | \( 1 + 6 T + p T^{2} \) |
| 31 | \( 1 - 8 T + p T^{2} \) |
| 37 | \( 1 - 10 T + p T^{2} \) |
| 41 | \( 1 - 10 T + p T^{2} \) |
| 43 | \( 1 + 12 T + p T^{2} \) |
| 47 | \( 1 - 8 T + p T^{2} \) |
| 53 | \( 1 + 6 T + p T^{2} \) |
| 59 | \( 1 - 4 T + p T^{2} \) |
| 61 | \( 1 + 10 T + p T^{2} \) |
| 67 | \( 1 + 12 T + p T^{2} \) |
| 71 | \( 1 - 4 T + p T^{2} \) |
| 73 | \( 1 + 2 T + p T^{2} \) |
| 79 | \( 1 - 8 T + p T^{2} \) |
| 83 | \( 1 - 4 T + p T^{2} \) |
| 89 | \( 1 + 6 T + p T^{2} \) |
| 97 | \( 1 + 10 T + p T^{2} \) |
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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
−7.50145721297911331954172165127, −6.37737416079205009402799745509, −6.31966226462749016090174821710, −5.29804958682390229091970978788, −4.41742451171843232672097083373, −4.00181781862111816036116911957, −2.68627617651818423094044515386, −2.33414740066152855700267422996, −1.49603267104550663371017102057, 0,
1.49603267104550663371017102057, 2.33414740066152855700267422996, 2.68627617651818423094044515386, 4.00181781862111816036116911957, 4.41742451171843232672097083373, 5.29804958682390229091970978788, 6.31966226462749016090174821710, 6.37737416079205009402799745509, 7.50145721297911331954172165127