| L(s) = 1 | + 2.30·3-s − 5-s + 1.53·7-s + 2.30·9-s + 2.44·11-s − 6.18·13-s − 2.30·15-s − 0.345·19-s + 3.52·21-s − 9.04·23-s + 25-s − 1.60·27-s − 5.05·29-s − 2.71·31-s + 5.63·33-s − 1.53·35-s − 4.08·37-s − 14.2·39-s + 1.14·41-s + 0.730·43-s − 2.30·45-s + 0.594·47-s − 4.65·49-s − 9.59·53-s − 2.44·55-s − 0.796·57-s + 6.74·59-s + ⋯ |
| L(s) = 1 | + 1.32·3-s − 0.447·5-s + 0.579·7-s + 0.767·9-s + 0.738·11-s − 1.71·13-s − 0.594·15-s − 0.0793·19-s + 0.769·21-s − 1.88·23-s + 0.200·25-s − 0.308·27-s − 0.938·29-s − 0.486·31-s + 0.981·33-s − 0.258·35-s − 0.671·37-s − 2.27·39-s + 0.178·41-s + 0.111·43-s − 0.343·45-s + 0.0866·47-s − 0.664·49-s − 1.31·53-s − 0.330·55-s − 0.105·57-s + 0.878·59-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5780 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5780 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 17 | \( 1 \) |
| good | 3 | \( 1 - 2.30T + 3T^{2} \) |
| 7 | \( 1 - 1.53T + 7T^{2} \) |
| 11 | \( 1 - 2.44T + 11T^{2} \) |
| 13 | \( 1 + 6.18T + 13T^{2} \) |
| 19 | \( 1 + 0.345T + 19T^{2} \) |
| 23 | \( 1 + 9.04T + 23T^{2} \) |
| 29 | \( 1 + 5.05T + 29T^{2} \) |
| 31 | \( 1 + 2.71T + 31T^{2} \) |
| 37 | \( 1 + 4.08T + 37T^{2} \) |
| 41 | \( 1 - 1.14T + 41T^{2} \) |
| 43 | \( 1 - 0.730T + 43T^{2} \) |
| 47 | \( 1 - 0.594T + 47T^{2} \) |
| 53 | \( 1 + 9.59T + 53T^{2} \) |
| 59 | \( 1 - 6.74T + 59T^{2} \) |
| 61 | \( 1 + 10.5T + 61T^{2} \) |
| 67 | \( 1 - 12.3T + 67T^{2} \) |
| 71 | \( 1 - 1.08T + 71T^{2} \) |
| 73 | \( 1 - 9.34T + 73T^{2} \) |
| 79 | \( 1 + 11.0T + 79T^{2} \) |
| 83 | \( 1 + 12.2T + 83T^{2} \) |
| 89 | \( 1 + 10.8T + 89T^{2} \) |
| 97 | \( 1 - 6.97T + 97T^{2} \) |
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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
−7.66232254640886816778880574140, −7.50055777015000893810961158593, −6.50992209411260711768812170601, −5.50906558811983721438424938680, −4.62524385179788998002805989767, −3.96106369925866521100592777187, −3.26959733955894637319246931256, −2.27179475262027607603328519774, −1.74367693689931642891740478092, 0,
1.74367693689931642891740478092, 2.27179475262027607603328519774, 3.26959733955894637319246931256, 3.96106369925866521100592777187, 4.62524385179788998002805989767, 5.50906558811983721438424938680, 6.50992209411260711768812170601, 7.50055777015000893810961158593, 7.66232254640886816778880574140
