L(s) = 1 | + 2-s + 4-s − 1.18·5-s + 0.532·7-s + 8-s − 1.18·10-s − 1.87·11-s − 3.87·13-s + 0.532·14-s + 16-s − 1.16·17-s − 1.18·20-s − 1.87·22-s + 6.70·23-s − 3.59·25-s − 3.87·26-s + 0.532·28-s + 4.02·29-s − 1.95·31-s + 32-s − 1.16·34-s − 0.630·35-s + 6.88·37-s − 1.18·40-s + 8.98·41-s + 2.42·43-s − 1.87·44-s + ⋯ |
L(s) = 1 | + 0.707·2-s + 0.5·4-s − 0.529·5-s + 0.201·7-s + 0.353·8-s − 0.374·10-s − 0.566·11-s − 1.07·13-s + 0.142·14-s + 0.250·16-s − 0.281·17-s − 0.264·20-s − 0.400·22-s + 1.39·23-s − 0.719·25-s − 0.760·26-s + 0.100·28-s + 0.746·29-s − 0.351·31-s + 0.176·32-s − 0.199·34-s − 0.106·35-s + 1.13·37-s − 0.187·40-s + 1.40·41-s + 0.369·43-s − 0.283·44-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6498 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6498 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 19 | \( 1 \) |
good | 5 | \( 1 + 1.18T + 5T^{2} \) |
| 7 | \( 1 - 0.532T + 7T^{2} \) |
| 11 | \( 1 + 1.87T + 11T^{2} \) |
| 13 | \( 1 + 3.87T + 13T^{2} \) |
| 17 | \( 1 + 1.16T + 17T^{2} \) |
| 23 | \( 1 - 6.70T + 23T^{2} \) |
| 29 | \( 1 - 4.02T + 29T^{2} \) |
| 31 | \( 1 + 1.95T + 31T^{2} \) |
| 37 | \( 1 - 6.88T + 37T^{2} \) |
| 41 | \( 1 - 8.98T + 41T^{2} \) |
| 43 | \( 1 - 2.42T + 43T^{2} \) |
| 47 | \( 1 + 2.04T + 47T^{2} \) |
| 53 | \( 1 + 12.9T + 53T^{2} \) |
| 59 | \( 1 - 2.68T + 59T^{2} \) |
| 61 | \( 1 + 11.3T + 61T^{2} \) |
| 67 | \( 1 + 11.1T + 67T^{2} \) |
| 71 | \( 1 + 6.07T + 71T^{2} \) |
| 73 | \( 1 + 0.327T + 73T^{2} \) |
| 79 | \( 1 + 16.1T + 79T^{2} \) |
| 83 | \( 1 + 11.5T + 83T^{2} \) |
| 89 | \( 1 - 3.55T + 89T^{2} \) |
| 97 | \( 1 - 5.87T + 97T^{2} \) |
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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
−7.63890508546975325272800666509, −7.00251330694094498739389166390, −6.17093943603903382065279336794, −5.41817710785434758896830764875, −4.62570176485452041504942707758, −4.28185144794084407930903773749, −3.05388853717721620588898350808, −2.64321926093916871477165433262, −1.43830702942725739162633898221, 0,
1.43830702942725739162633898221, 2.64321926093916871477165433262, 3.05388853717721620588898350808, 4.28185144794084407930903773749, 4.62570176485452041504942707758, 5.41817710785434758896830764875, 6.17093943603903382065279336794, 7.00251330694094498739389166390, 7.63890508546975325272800666509