| L(s) = 1 | − 3.41·5-s − 2·11-s + 2.58·13-s + 2.24·17-s + 2.82·19-s − 7.65·23-s + 6.65·25-s + 6.82·29-s + 1.17·31-s − 4·37-s − 6.24·41-s − 5.65·43-s − 2.82·47-s + 2·53-s + 6.82·55-s − 1.17·59-s + 12.2·61-s − 8.82·65-s + 5.65·67-s + 9.31·71-s + 13.8·73-s − 13.6·79-s + 7.31·83-s − 7.65·85-s + 14.2·89-s − 9.65·95-s + 2.58·97-s + ⋯ |
| L(s) = 1 | − 1.52·5-s − 0.603·11-s + 0.717·13-s + 0.543·17-s + 0.648·19-s − 1.59·23-s + 1.33·25-s + 1.26·29-s + 0.210·31-s − 0.657·37-s − 0.974·41-s − 0.862·43-s − 0.412·47-s + 0.274·53-s + 0.920·55-s − 0.152·59-s + 1.56·61-s − 1.09·65-s + 0.691·67-s + 1.10·71-s + 1.62·73-s − 1.53·79-s + 0.802·83-s − 0.830·85-s + 1.50·89-s − 0.990·95-s + 0.262·97-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
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 7 | \( 1 \) |
| good | 5 | \( 1 + 3.41T + 5T^{2} \) |
| 11 | \( 1 + 2T + 11T^{2} \) |
| 13 | \( 1 - 2.58T + 13T^{2} \) |
| 17 | \( 1 - 2.24T + 17T^{2} \) |
| 19 | \( 1 - 2.82T + 19T^{2} \) |
| 23 | \( 1 + 7.65T + 23T^{2} \) |
| 29 | \( 1 - 6.82T + 29T^{2} \) |
| 31 | \( 1 - 1.17T + 31T^{2} \) |
| 37 | \( 1 + 4T + 37T^{2} \) |
| 41 | \( 1 + 6.24T + 41T^{2} \) |
| 43 | \( 1 + 5.65T + 43T^{2} \) |
| 47 | \( 1 + 2.82T + 47T^{2} \) |
| 53 | \( 1 - 2T + 53T^{2} \) |
| 59 | \( 1 + 1.17T + 59T^{2} \) |
| 61 | \( 1 - 12.2T + 61T^{2} \) |
| 67 | \( 1 - 5.65T + 67T^{2} \) |
| 71 | \( 1 - 9.31T + 71T^{2} \) |
| 73 | \( 1 - 13.8T + 73T^{2} \) |
| 79 | \( 1 + 13.6T + 79T^{2} \) |
| 83 | \( 1 - 7.31T + 83T^{2} \) |
| 89 | \( 1 - 14.2T + 89T^{2} \) |
| 97 | \( 1 - 2.58T + 97T^{2} \) |
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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
−7.78448862739273330815218123613, −6.96829823455528476656538533109, −6.31669332329476244176907301385, −5.33075773874890257026782941947, −4.73314714532105852460388231046, −3.70208936368141272254606387191, −3.50482297221063585092269416635, −2.37022443881284216851369091798, −1.08560958480826197450945713323, 0,
1.08560958480826197450945713323, 2.37022443881284216851369091798, 3.50482297221063585092269416635, 3.70208936368141272254606387191, 4.73314714532105852460388231046, 5.33075773874890257026782941947, 6.31669332329476244176907301385, 6.96829823455528476656538533109, 7.78448862739273330815218123613
