| L(s) = 1 | − 2-s + 4-s + 1.77·5-s − 1.25·7-s − 8-s − 1.77·10-s + 2.17·11-s + 0.350·13-s + 1.25·14-s + 16-s + 0.495·17-s − 2.51·19-s + 1.77·20-s − 2.17·22-s − 1.85·25-s − 0.350·26-s − 1.25·28-s − 0.774·29-s + 4.19·31-s − 32-s − 0.495·34-s − 2.22·35-s + 2.77·37-s + 2.51·38-s − 1.77·40-s − 8.15·41-s − 10.0·43-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 0.5·4-s + 0.792·5-s − 0.474·7-s − 0.353·8-s − 0.560·10-s + 0.655·11-s + 0.0971·13-s + 0.335·14-s + 0.250·16-s + 0.120·17-s − 0.575·19-s + 0.396·20-s − 0.463·22-s − 0.371·25-s − 0.0687·26-s − 0.237·28-s − 0.143·29-s + 0.754·31-s − 0.176·32-s − 0.0849·34-s − 0.376·35-s + 0.455·37-s + 0.407·38-s − 0.280·40-s − 1.27·41-s − 1.52·43-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 9522 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 9522 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 3 | \( 1 \) |
| 23 | \( 1 \) |
| good | 5 | \( 1 - 1.77T + 5T^{2} \) |
| 7 | \( 1 + 1.25T + 7T^{2} \) |
| 11 | \( 1 - 2.17T + 11T^{2} \) |
| 13 | \( 1 - 0.350T + 13T^{2} \) |
| 17 | \( 1 - 0.495T + 17T^{2} \) |
| 19 | \( 1 + 2.51T + 19T^{2} \) |
| 29 | \( 1 + 0.774T + 29T^{2} \) |
| 31 | \( 1 - 4.19T + 31T^{2} \) |
| 37 | \( 1 - 2.77T + 37T^{2} \) |
| 41 | \( 1 + 8.15T + 41T^{2} \) |
| 43 | \( 1 + 10.0T + 43T^{2} \) |
| 47 | \( 1 - 7.77T + 47T^{2} \) |
| 53 | \( 1 + 5.99T + 53T^{2} \) |
| 59 | \( 1 + 3.35T + 59T^{2} \) |
| 61 | \( 1 - 8.58T + 61T^{2} \) |
| 67 | \( 1 + 4.58T + 67T^{2} \) |
| 71 | \( 1 + 14.4T + 71T^{2} \) |
| 73 | \( 1 - 10.1T + 73T^{2} \) |
| 79 | \( 1 + 11.0T + 79T^{2} \) |
| 83 | \( 1 + 4.91T + 83T^{2} \) |
| 89 | \( 1 - 10.4T + 89T^{2} \) |
| 97 | \( 1 - 0.674T + 97T^{2} \) |
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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
−7.32611546626913928804971159176, −6.57942056981766192168309251159, −6.23675022035651337637419618773, −5.49903011185219681019603351692, −4.59398812741473215520165548303, −3.67558514451846344764343532357, −2.88990372677529900510762076423, −1.98986413176447260101797841218, −1.28264157129613450745864763285, 0,
1.28264157129613450745864763285, 1.98986413176447260101797841218, 2.88990372677529900510762076423, 3.67558514451846344764343532357, 4.59398812741473215520165548303, 5.49903011185219681019603351692, 6.23675022035651337637419618773, 6.57942056981766192168309251159, 7.32611546626913928804971159176
