| L(s) = 1 | − 2.82·5-s − 3·9-s + 11-s − 2.82·17-s + 2.82·19-s + 3.00·25-s + 2·29-s + 8.48·31-s − 6·37-s + 8.48·41-s − 4·43-s + 8.48·45-s + 2.82·47-s + 6·53-s − 2.82·55-s + 5.65·59-s − 4·67-s − 8·71-s − 2.82·73-s + 9·81-s + 14.1·83-s + 8.00·85-s − 11.3·89-s − 8.00·95-s − 3·99-s + 5.65·101-s − 8.48·103-s + ⋯ |
| L(s) = 1 | − 1.26·5-s − 9-s + 0.301·11-s − 0.685·17-s + 0.648·19-s + 0.600·25-s + 0.371·29-s + 1.52·31-s − 0.986·37-s + 1.32·41-s − 0.609·43-s + 1.26·45-s + 0.412·47-s + 0.824·53-s − 0.381·55-s + 0.736·59-s − 0.488·67-s − 0.949·71-s − 0.331·73-s + 81-s + 1.55·83-s + 0.867·85-s − 1.19·89-s − 0.820·95-s − 0.301·99-s + 0.562·101-s − 0.836·103-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8624 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8624 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 7 | \( 1 \) |
| 11 | \( 1 - T \) |
| good | 3 | \( 1 + 3T^{2} \) |
| 5 | \( 1 + 2.82T + 5T^{2} \) |
| 13 | \( 1 + 13T^{2} \) |
| 17 | \( 1 + 2.82T + 17T^{2} \) |
| 19 | \( 1 - 2.82T + 19T^{2} \) |
| 23 | \( 1 + 23T^{2} \) |
| 29 | \( 1 - 2T + 29T^{2} \) |
| 31 | \( 1 - 8.48T + 31T^{2} \) |
| 37 | \( 1 + 6T + 37T^{2} \) |
| 41 | \( 1 - 8.48T + 41T^{2} \) |
| 43 | \( 1 + 4T + 43T^{2} \) |
| 47 | \( 1 - 2.82T + 47T^{2} \) |
| 53 | \( 1 - 6T + 53T^{2} \) |
| 59 | \( 1 - 5.65T + 59T^{2} \) |
| 61 | \( 1 + 61T^{2} \) |
| 67 | \( 1 + 4T + 67T^{2} \) |
| 71 | \( 1 + 8T + 71T^{2} \) |
| 73 | \( 1 + 2.82T + 73T^{2} \) |
| 79 | \( 1 + 79T^{2} \) |
| 83 | \( 1 - 14.1T + 83T^{2} \) |
| 89 | \( 1 + 11.3T + 89T^{2} \) |
| 97 | \( 1 + 97T^{2} \) |
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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
−7.49662034113061995714737166143, −6.81445739331860437939291185897, −6.11062722492915597901931273792, −5.30092549669356077297913184979, −4.52368041144072138819011285022, −3.88043418444146723364950339801, −3.12141312940782717125120136486, −2.40559530268297153961283622607, −1.01739207421506705941148561256, 0,
1.01739207421506705941148561256, 2.40559530268297153961283622607, 3.12141312940782717125120136486, 3.88043418444146723364950339801, 4.52368041144072138819011285022, 5.30092549669356077297913184979, 6.11062722492915597901931273792, 6.81445739331860437939291185897, 7.49662034113061995714737166143
