| L(s) = 1 | + 4.27·5-s − 4.27·11-s − 1.27·13-s + 4·17-s − 1.27·19-s + 4·23-s + 13.2·25-s + 2.27·29-s + 31-s + 5.27·37-s − 10.5·41-s + 7.27·43-s − 6·47-s − 1.72·53-s − 18.2·55-s + 6.27·59-s + 10·61-s − 5.45·65-s − 7.27·67-s + 2·71-s + 3.27·73-s + 3.54·79-s − 0.274·83-s + 17.0·85-s − 4.54·89-s − 5.45·95-s + 16.2·97-s + ⋯ |
| L(s) = 1 | + 1.91·5-s − 1.28·11-s − 0.353·13-s + 0.970·17-s − 0.292·19-s + 0.834·23-s + 2.65·25-s + 0.422·29-s + 0.179·31-s + 0.867·37-s − 1.64·41-s + 1.10·43-s − 0.875·47-s − 0.236·53-s − 2.46·55-s + 0.816·59-s + 1.28·61-s − 0.676·65-s − 0.888·67-s + 0.237·71-s + 0.383·73-s + 0.399·79-s − 0.0301·83-s + 1.85·85-s − 0.482·89-s − 0.559·95-s + 1.65·97-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 7056 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7056 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.959448635\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.959448635\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 7 | \( 1 \) |
| good | 5 | \( 1 - 4.27T + 5T^{2} \) |
| 11 | \( 1 + 4.27T + 11T^{2} \) |
| 13 | \( 1 + 1.27T + 13T^{2} \) |
| 17 | \( 1 - 4T + 17T^{2} \) |
| 19 | \( 1 + 1.27T + 19T^{2} \) |
| 23 | \( 1 - 4T + 23T^{2} \) |
| 29 | \( 1 - 2.27T + 29T^{2} \) |
| 31 | \( 1 - T + 31T^{2} \) |
| 37 | \( 1 - 5.27T + 37T^{2} \) |
| 41 | \( 1 + 10.5T + 41T^{2} \) |
| 43 | \( 1 - 7.27T + 43T^{2} \) |
| 47 | \( 1 + 6T + 47T^{2} \) |
| 53 | \( 1 + 1.72T + 53T^{2} \) |
| 59 | \( 1 - 6.27T + 59T^{2} \) |
| 61 | \( 1 - 10T + 61T^{2} \) |
| 67 | \( 1 + 7.27T + 67T^{2} \) |
| 71 | \( 1 - 2T + 71T^{2} \) |
| 73 | \( 1 - 3.27T + 73T^{2} \) |
| 79 | \( 1 - 3.54T + 79T^{2} \) |
| 83 | \( 1 + 0.274T + 83T^{2} \) |
| 89 | \( 1 + 4.54T + 89T^{2} \) |
| 97 | \( 1 - 16.2T + 97T^{2} \) |
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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
−7.978972541711201537128547440650, −7.14409268122109976167282740627, −6.44923695614166910188326298781, −5.74919061410498442656330597383, −5.21640717181545932998557282690, −4.70738293764577312063480169105, −3.27123620528034152938493649687, −2.60512890550146842939370464130, −1.93561795158110418724727232009, −0.888183035665950526822857826380,
0.888183035665950526822857826380, 1.93561795158110418724727232009, 2.60512890550146842939370464130, 3.27123620528034152938493649687, 4.70738293764577312063480169105, 5.21640717181545932998557282690, 5.74919061410498442656330597383, 6.44923695614166910188326298781, 7.14409268122109976167282740627, 7.978972541711201537128547440650
