| L(s) = 1 | + 1.94·2-s + 3-s + 1.80·4-s + 1.88·5-s + 1.94·6-s − 7-s − 0.388·8-s + 9-s + 3.67·10-s − 4.90·11-s + 1.80·12-s + 2.08·13-s − 1.94·14-s + 1.88·15-s − 4.35·16-s − 6.37·17-s + 1.94·18-s + 0.978·19-s + 3.39·20-s − 21-s − 9.56·22-s − 5.35·23-s − 0.388·24-s − 1.44·25-s + 4.05·26-s + 27-s − 1.80·28-s + ⋯ |
| L(s) = 1 | + 1.37·2-s + 0.577·3-s + 0.900·4-s + 0.843·5-s + 0.795·6-s − 0.377·7-s − 0.137·8-s + 0.333·9-s + 1.16·10-s − 1.47·11-s + 0.519·12-s + 0.577·13-s − 0.521·14-s + 0.486·15-s − 1.08·16-s − 1.54·17-s + 0.459·18-s + 0.224·19-s + 0.759·20-s − 0.218·21-s − 2.03·22-s − 1.11·23-s − 0.0793·24-s − 0.288·25-s + 0.795·26-s + 0.192·27-s − 0.340·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8043 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8043 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) |
| 7 | \( 1 + T \) |
| 383 | \( 1 - T \) |
| good | 2 | \( 1 - 1.94T + 2T^{2} \) |
| 5 | \( 1 - 1.88T + 5T^{2} \) |
| 11 | \( 1 + 4.90T + 11T^{2} \) |
| 13 | \( 1 - 2.08T + 13T^{2} \) |
| 17 | \( 1 + 6.37T + 17T^{2} \) |
| 19 | \( 1 - 0.978T + 19T^{2} \) |
| 23 | \( 1 + 5.35T + 23T^{2} \) |
| 29 | \( 1 - 4.49T + 29T^{2} \) |
| 31 | \( 1 + 7.52T + 31T^{2} \) |
| 37 | \( 1 - 1.28T + 37T^{2} \) |
| 41 | \( 1 - 6.41T + 41T^{2} \) |
| 43 | \( 1 - 2.44T + 43T^{2} \) |
| 47 | \( 1 - 3.01T + 47T^{2} \) |
| 53 | \( 1 + 14.1T + 53T^{2} \) |
| 59 | \( 1 - 6.83T + 59T^{2} \) |
| 61 | \( 1 + 6.25T + 61T^{2} \) |
| 67 | \( 1 - 1.90T + 67T^{2} \) |
| 71 | \( 1 + 14.1T + 71T^{2} \) |
| 73 | \( 1 + 10.0T + 73T^{2} \) |
| 79 | \( 1 - 4.08T + 79T^{2} \) |
| 83 | \( 1 + 1.63T + 83T^{2} \) |
| 89 | \( 1 + 1.10T + 89T^{2} \) |
| 97 | \( 1 - 8.28T + 97T^{2} \) |
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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
−7.36357381629875890124546449828, −6.46263864886141234729004693079, −5.97693561675614468795522774922, −5.40798757236915618143985308939, −4.57198593884353629899423463495, −4.00848352096740702156003769237, −3.08198571576383672972807039753, −2.49372131506739208236432194638, −1.85605843692422623105966020846, 0,
1.85605843692422623105966020846, 2.49372131506739208236432194638, 3.08198571576383672972807039753, 4.00848352096740702156003769237, 4.57198593884353629899423463495, 5.40798757236915618143985308939, 5.97693561675614468795522774922, 6.46263864886141234729004693079, 7.36357381629875890124546449828
