| L(s) = 1 | + 2.11·2-s − 3-s + 2.45·4-s − 2.11·6-s − 2.18·7-s + 0.961·8-s + 9-s + 2.72·11-s − 2.45·12-s − 6.21·13-s − 4.60·14-s − 2.88·16-s + 0.494·17-s + 2.11·18-s + 8.10·19-s + 2.18·21-s + 5.74·22-s + 1.62·23-s − 0.961·24-s − 13.1·26-s − 27-s − 5.36·28-s − 1.55·29-s + 0.301·31-s − 8.00·32-s − 2.72·33-s + 1.04·34-s + ⋯ |
| L(s) = 1 | + 1.49·2-s − 0.577·3-s + 1.22·4-s − 0.861·6-s − 0.825·7-s + 0.340·8-s + 0.333·9-s + 0.821·11-s − 0.708·12-s − 1.72·13-s − 1.23·14-s − 0.720·16-s + 0.119·17-s + 0.497·18-s + 1.85·19-s + 0.476·21-s + 1.22·22-s + 0.339·23-s − 0.196·24-s − 2.57·26-s − 0.192·27-s − 1.01·28-s − 0.289·29-s + 0.0542·31-s − 1.41·32-s − 0.474·33-s + 0.179·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8025 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8025 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 3 | \( 1 + T \) |
| 5 | \( 1 \) |
| 107 | \( 1 + T \) |
| good | 2 | \( 1 - 2.11T + 2T^{2} \) |
| 7 | \( 1 + 2.18T + 7T^{2} \) |
| 11 | \( 1 - 2.72T + 11T^{2} \) |
| 13 | \( 1 + 6.21T + 13T^{2} \) |
| 17 | \( 1 - 0.494T + 17T^{2} \) |
| 19 | \( 1 - 8.10T + 19T^{2} \) |
| 23 | \( 1 - 1.62T + 23T^{2} \) |
| 29 | \( 1 + 1.55T + 29T^{2} \) |
| 31 | \( 1 - 0.301T + 31T^{2} \) |
| 37 | \( 1 - 10.8T + 37T^{2} \) |
| 41 | \( 1 - 4.60T + 41T^{2} \) |
| 43 | \( 1 + 4.77T + 43T^{2} \) |
| 47 | \( 1 - 6.82T + 47T^{2} \) |
| 53 | \( 1 + 8.36T + 53T^{2} \) |
| 59 | \( 1 + 13.9T + 59T^{2} \) |
| 61 | \( 1 + 4.89T + 61T^{2} \) |
| 67 | \( 1 + 5.35T + 67T^{2} \) |
| 71 | \( 1 + 16.7T + 71T^{2} \) |
| 73 | \( 1 + 10.7T + 73T^{2} \) |
| 79 | \( 1 + 7.47T + 79T^{2} \) |
| 83 | \( 1 - 13.5T + 83T^{2} \) |
| 89 | \( 1 + 1.42T + 89T^{2} \) |
| 97 | \( 1 + 13.8T + 97T^{2} \) |
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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
−7.31878378163705480472411776780, −6.50752276548292065441378225362, −5.98186416871545802335447179610, −5.33492570670345898297175928862, −4.66776416841695188986584339354, −4.12350917024419134837128755314, −3.08544923929999003300578989943, −2.77646641223030008089236609735, −1.39500841203897567169093052493, 0,
1.39500841203897567169093052493, 2.77646641223030008089236609735, 3.08544923929999003300578989943, 4.12350917024419134837128755314, 4.66776416841695188986584339354, 5.33492570670345898297175928862, 5.98186416871545802335447179610, 6.50752276548292065441378225362, 7.31878378163705480472411776780
