| L(s) = 1 | + 3-s − 1.50·7-s + 9-s − 6.17·11-s + 3.55·13-s − 0.495·17-s + 0.311·19-s − 1.50·21-s + 3.06·23-s + 27-s − 0.122·29-s − 2.94·31-s − 6.17·33-s + 4.36·37-s + 3.55·39-s + 4.25·41-s + 3.62·43-s − 5.28·47-s − 4.73·49-s − 0.495·51-s + 8.59·53-s + 0.311·57-s + 12.8·59-s − 11.3·61-s − 1.50·63-s − 13.2·67-s + 3.06·69-s + ⋯ |
| L(s) = 1 | + 0.577·3-s − 0.568·7-s + 0.333·9-s − 1.86·11-s + 0.986·13-s − 0.120·17-s + 0.0715·19-s − 0.328·21-s + 0.638·23-s + 0.192·27-s − 0.0227·29-s − 0.528·31-s − 1.07·33-s + 0.717·37-s + 0.569·39-s + 0.664·41-s + 0.553·43-s − 0.770·47-s − 0.676·49-s − 0.0694·51-s + 1.18·53-s + 0.0412·57-s + 1.67·59-s − 1.44·61-s − 0.189·63-s − 1.61·67-s + 0.368·69-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 7500 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7500 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) |
| 5 | \( 1 \) |
| good | 7 | \( 1 + 1.50T + 7T^{2} \) |
| 11 | \( 1 + 6.17T + 11T^{2} \) |
| 13 | \( 1 - 3.55T + 13T^{2} \) |
| 17 | \( 1 + 0.495T + 17T^{2} \) |
| 19 | \( 1 - 0.311T + 19T^{2} \) |
| 23 | \( 1 - 3.06T + 23T^{2} \) |
| 29 | \( 1 + 0.122T + 29T^{2} \) |
| 31 | \( 1 + 2.94T + 31T^{2} \) |
| 37 | \( 1 - 4.36T + 37T^{2} \) |
| 41 | \( 1 - 4.25T + 41T^{2} \) |
| 43 | \( 1 - 3.62T + 43T^{2} \) |
| 47 | \( 1 + 5.28T + 47T^{2} \) |
| 53 | \( 1 - 8.59T + 53T^{2} \) |
| 59 | \( 1 - 12.8T + 59T^{2} \) |
| 61 | \( 1 + 11.3T + 61T^{2} \) |
| 67 | \( 1 + 13.2T + 67T^{2} \) |
| 71 | \( 1 - 3.26T + 71T^{2} \) |
| 73 | \( 1 + 15.9T + 73T^{2} \) |
| 79 | \( 1 + 8.52T + 79T^{2} \) |
| 83 | \( 1 + 11.3T + 83T^{2} \) |
| 89 | \( 1 - 14.1T + 89T^{2} \) |
| 97 | \( 1 - 11.4T + 97T^{2} \) |
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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
−7.58706328498624684442384779019, −7.01528071560361279127385627588, −6.07056121444824060138818000121, −5.51737162701458184492985599283, −4.66051484926470341592122649230, −3.83137183545852935848772300452, −2.96869992917809156462200744161, −2.52790879814028245349407148521, −1.32266362859485304970422989729, 0,
1.32266362859485304970422989729, 2.52790879814028245349407148521, 2.96869992917809156462200744161, 3.83137183545852935848772300452, 4.66051484926470341592122649230, 5.51737162701458184492985599283, 6.07056121444824060138818000121, 7.01528071560361279127385627588, 7.58706328498624684442384779019
