| L(s) = 1 | − 2-s + 2.79·3-s + 4-s + 4.02·5-s − 2.79·6-s − 0.274·7-s − 8-s + 4.79·9-s − 4.02·10-s + 2.79·12-s − 0.597·13-s + 0.274·14-s + 11.2·15-s + 16-s + 2.62·17-s − 4.79·18-s + 19-s + 4.02·20-s − 0.767·21-s + 4.04·23-s − 2.79·24-s + 11.1·25-s + 0.597·26-s + 4.99·27-s − 0.274·28-s + 7.29·29-s − 11.2·30-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 1.61·3-s + 0.5·4-s + 1.79·5-s − 1.13·6-s − 0.103·7-s − 0.353·8-s + 1.59·9-s − 1.27·10-s + 0.805·12-s − 0.165·13-s + 0.0734·14-s + 2.89·15-s + 0.250·16-s + 0.637·17-s − 1.12·18-s + 0.229·19-s + 0.899·20-s − 0.167·21-s + 0.843·23-s − 0.569·24-s + 2.23·25-s + 0.117·26-s + 0.962·27-s − 0.0519·28-s + 1.35·29-s − 2.05·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4598 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4598 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.887088301\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.887088301\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + T \) |
| 11 | \( 1 \) |
| 19 | \( 1 - T \) |
| good | 3 | \( 1 - 2.79T + 3T^{2} \) |
| 5 | \( 1 - 4.02T + 5T^{2} \) |
| 7 | \( 1 + 0.274T + 7T^{2} \) |
| 13 | \( 1 + 0.597T + 13T^{2} \) |
| 17 | \( 1 - 2.62T + 17T^{2} \) |
| 23 | \( 1 - 4.04T + 23T^{2} \) |
| 29 | \( 1 - 7.29T + 29T^{2} \) |
| 31 | \( 1 + 1.48T + 31T^{2} \) |
| 37 | \( 1 + 8.31T + 37T^{2} \) |
| 41 | \( 1 - 12.3T + 41T^{2} \) |
| 43 | \( 1 + 5.58T + 43T^{2} \) |
| 47 | \( 1 + 9.10T + 47T^{2} \) |
| 53 | \( 1 - 7.30T + 53T^{2} \) |
| 59 | \( 1 + 7.53T + 59T^{2} \) |
| 61 | \( 1 + 4.83T + 61T^{2} \) |
| 67 | \( 1 + 14.0T + 67T^{2} \) |
| 71 | \( 1 + 9.61T + 71T^{2} \) |
| 73 | \( 1 - 3.15T + 73T^{2} \) |
| 79 | \( 1 - 2.42T + 79T^{2} \) |
| 83 | \( 1 + 15.7T + 83T^{2} \) |
| 89 | \( 1 - 1.49T + 89T^{2} \) |
| 97 | \( 1 - 9.00T + 97T^{2} \) |
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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
−8.542059510862506552252505549477, −7.76643546484801317608759687461, −7.03375446565863422671825031967, −6.33602629270980713671825421091, −5.50160891405612213977670471675, −4.59355905372068623433634471182, −3.16725376968171598884815843166, −2.85062265538887166126993282975, −1.89875889866990554619032283438, −1.28502238627651904527915799619,
1.28502238627651904527915799619, 1.89875889866990554619032283438, 2.85062265538887166126993282975, 3.16725376968171598884815843166, 4.59355905372068623433634471182, 5.50160891405612213977670471675, 6.33602629270980713671825421091, 7.03375446565863422671825031967, 7.76643546484801317608759687461, 8.542059510862506552252505549477
