L(s) = 1 | − 0.523·3-s + 1.52·5-s + 4.20·7-s − 2.72·9-s + 4.67·11-s + 6.72·13-s − 0.798·15-s + 6.40·17-s + 2·19-s − 2.20·21-s − 2.72·23-s − 2.67·25-s + 3·27-s − 1.67·29-s − 10.9·31-s − 2.45·33-s + 6.40·35-s + 37-s − 3.52·39-s + 4.52·41-s − 3.45·43-s − 4.15·45-s − 6.20·47-s + 10.6·49-s − 3.35·51-s − 3.24·53-s + 7.12·55-s + ⋯ |
L(s) = 1 | − 0.302·3-s + 0.681·5-s + 1.58·7-s − 0.908·9-s + 1.41·11-s + 1.86·13-s − 0.206·15-s + 1.55·17-s + 0.458·19-s − 0.480·21-s − 0.568·23-s − 0.535·25-s + 0.577·27-s − 0.311·29-s − 1.97·31-s − 0.426·33-s + 1.08·35-s + 0.164·37-s − 0.564·39-s + 0.706·41-s − 0.526·43-s − 0.619·45-s − 0.904·47-s + 1.52·49-s − 0.469·51-s − 0.446·53-s + 0.961·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2368 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2368 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.579855230\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.579855230\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 37 | \( 1 - T \) |
good | 3 | \( 1 + 0.523T + 3T^{2} \) |
| 5 | \( 1 - 1.52T + 5T^{2} \) |
| 7 | \( 1 - 4.20T + 7T^{2} \) |
| 11 | \( 1 - 4.67T + 11T^{2} \) |
| 13 | \( 1 - 6.72T + 13T^{2} \) |
| 17 | \( 1 - 6.40T + 17T^{2} \) |
| 19 | \( 1 - 2T + 19T^{2} \) |
| 23 | \( 1 + 2.72T + 23T^{2} \) |
| 29 | \( 1 + 1.67T + 29T^{2} \) |
| 31 | \( 1 + 10.9T + 31T^{2} \) |
| 41 | \( 1 - 4.52T + 41T^{2} \) |
| 43 | \( 1 + 3.45T + 43T^{2} \) |
| 47 | \( 1 + 6.20T + 47T^{2} \) |
| 53 | \( 1 + 3.24T + 53T^{2} \) |
| 59 | \( 1 + 9.85T + 59T^{2} \) |
| 61 | \( 1 - 6.87T + 61T^{2} \) |
| 67 | \( 1 - 5.52T + 67T^{2} \) |
| 71 | \( 1 + 5.15T + 71T^{2} \) |
| 73 | \( 1 + 7.62T + 73T^{2} \) |
| 79 | \( 1 + 2.32T + 79T^{2} \) |
| 83 | \( 1 + 3.15T + 83T^{2} \) |
| 89 | \( 1 - 16.4T + 89T^{2} \) |
| 97 | \( 1 + 10.8T + 97T^{2} \) |
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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
−8.942319773719463191180250939424, −8.242826661990157804677226062726, −7.60996652070868137832370812346, −6.40237392926557570668138628333, −5.74331234247448961634298716634, −5.32886023908098888610634837709, −4.07653466974140940160273971518, −3.35834248055579446704532694264, −1.75553888293726639985639260709, −1.25037454441759616459893870562,
1.25037454441759616459893870562, 1.75553888293726639985639260709, 3.35834248055579446704532694264, 4.07653466974140940160273971518, 5.32886023908098888610634837709, 5.74331234247448961634298716634, 6.40237392926557570668138628333, 7.60996652070868137832370812346, 8.242826661990157804677226062726, 8.942319773719463191180250939424