| L(s) = 1 | − 2-s + 4-s − 8-s + 4.24·11-s − 2.24·13-s + 16-s − 2.24·19-s − 4.24·22-s + 1.24·23-s − 5·25-s + 2.24·26-s − 4.24·29-s + 9.24·31-s − 32-s − 4·37-s + 2.24·38-s − 11.4·41-s + 10.4·43-s + 4.24·44-s − 1.24·46-s − 4.75·47-s + 5·50-s − 2.24·52-s + 4.24·53-s + 4.24·58-s − 2.24·61-s − 9.24·62-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 0.5·4-s − 0.353·8-s + 1.27·11-s − 0.621·13-s + 0.250·16-s − 0.514·19-s − 0.904·22-s + 0.259·23-s − 25-s + 0.439·26-s − 0.787·29-s + 1.66·31-s − 0.176·32-s − 0.657·37-s + 0.363·38-s − 1.79·41-s + 1.59·43-s + 0.639·44-s − 0.183·46-s − 0.693·47-s + 0.707·50-s − 0.310·52-s + 0.582·53-s + 0.557·58-s − 0.287·61-s − 1.17·62-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 7938 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7938 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 7 | \( 1 \) |
| good | 5 | \( 1 + 5T^{2} \) |
| 11 | \( 1 - 4.24T + 11T^{2} \) |
| 13 | \( 1 + 2.24T + 13T^{2} \) |
| 17 | \( 1 + 17T^{2} \) |
| 19 | \( 1 + 2.24T + 19T^{2} \) |
| 23 | \( 1 - 1.24T + 23T^{2} \) |
| 29 | \( 1 + 4.24T + 29T^{2} \) |
| 31 | \( 1 - 9.24T + 31T^{2} \) |
| 37 | \( 1 + 4T + 37T^{2} \) |
| 41 | \( 1 + 11.4T + 41T^{2} \) |
| 43 | \( 1 - 10.4T + 43T^{2} \) |
| 47 | \( 1 + 4.75T + 47T^{2} \) |
| 53 | \( 1 - 4.24T + 53T^{2} \) |
| 59 | \( 1 + 59T^{2} \) |
| 61 | \( 1 + 2.24T + 61T^{2} \) |
| 67 | \( 1 - 0.242T + 67T^{2} \) |
| 71 | \( 1 + 1.24T + 71T^{2} \) |
| 73 | \( 1 + 7T + 73T^{2} \) |
| 79 | \( 1 - 0.757T + 79T^{2} \) |
| 83 | \( 1 + 16.2T + 83T^{2} \) |
| 89 | \( 1 + 11.4T + 89T^{2} \) |
| 97 | \( 1 - 4.48T + 97T^{2} \) |
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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
−7.44547157283786570339904366446, −6.92763269346115620758097124197, −6.26931087984648627936420138201, −5.58225875109505807275131568194, −4.57683945792974113373398514337, −3.89933037120271914288259442941, −2.98821678416405042379509172268, −2.04407615467767609079328752608, −1.24461894250784106165013342370, 0,
1.24461894250784106165013342370, 2.04407615467767609079328752608, 2.98821678416405042379509172268, 3.89933037120271914288259442941, 4.57683945792974113373398514337, 5.58225875109505807275131568194, 6.26931087984648627936420138201, 6.92763269346115620758097124197, 7.44547157283786570339904366446
