| L(s) = 1 | + 3.86·3-s − 6.90·5-s − 7·7-s − 12.0·9-s − 30.9·11-s + 53.9·13-s − 26.6·15-s + 116.·17-s − 2.79·19-s − 27.0·21-s + 118.·23-s − 77.3·25-s − 150.·27-s − 112.·29-s − 213.·31-s − 119.·33-s + 48.3·35-s − 428.·37-s + 208.·39-s + 41·41-s + 457.·43-s + 83.3·45-s + 520.·47-s + 49·49-s + 449.·51-s + 133.·53-s + 213.·55-s + ⋯ |
| L(s) = 1 | + 0.743·3-s − 0.617·5-s − 0.377·7-s − 0.447·9-s − 0.847·11-s + 1.15·13-s − 0.459·15-s + 1.65·17-s − 0.0337·19-s − 0.281·21-s + 1.07·23-s − 0.618·25-s − 1.07·27-s − 0.717·29-s − 1.23·31-s − 0.629·33-s + 0.233·35-s − 1.90·37-s + 0.855·39-s + 0.156·41-s + 1.62·43-s + 0.275·45-s + 1.61·47-s + 0.142·49-s + 1.23·51-s + 0.346·53-s + 0.522·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1148 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1148 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(2.034627133\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.034627133\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 7 | \( 1 + 7T \) |
| 41 | \( 1 - 41T \) |
| good | 3 | \( 1 - 3.86T + 27T^{2} \) |
| 5 | \( 1 + 6.90T + 125T^{2} \) |
| 11 | \( 1 + 30.9T + 1.33e3T^{2} \) |
| 13 | \( 1 - 53.9T + 2.19e3T^{2} \) |
| 17 | \( 1 - 116.T + 4.91e3T^{2} \) |
| 19 | \( 1 + 2.79T + 6.85e3T^{2} \) |
| 23 | \( 1 - 118.T + 1.21e4T^{2} \) |
| 29 | \( 1 + 112.T + 2.43e4T^{2} \) |
| 31 | \( 1 + 213.T + 2.97e4T^{2} \) |
| 37 | \( 1 + 428.T + 5.06e4T^{2} \) |
| 43 | \( 1 - 457.T + 7.95e4T^{2} \) |
| 47 | \( 1 - 520.T + 1.03e5T^{2} \) |
| 53 | \( 1 - 133.T + 1.48e5T^{2} \) |
| 59 | \( 1 - 44.0T + 2.05e5T^{2} \) |
| 61 | \( 1 - 836.T + 2.26e5T^{2} \) |
| 67 | \( 1 - 274.T + 3.00e5T^{2} \) |
| 71 | \( 1 - 213.T + 3.57e5T^{2} \) |
| 73 | \( 1 + 132.T + 3.89e5T^{2} \) |
| 79 | \( 1 + 398.T + 4.93e5T^{2} \) |
| 83 | \( 1 - 1.11e3T + 5.71e5T^{2} \) |
| 89 | \( 1 - 1.21e3T + 7.04e5T^{2} \) |
| 97 | \( 1 - 993.T + 9.12e5T^{2} \) |
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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
−9.169312835666628577280917056470, −8.669076095804759980027435889706, −7.74336822653381720159826594241, −7.30541389423712184350038111377, −5.88758626226544082082133381731, −5.31048458901332118936358948508, −3.68408758135107676473088436769, −3.42783039718848135226359996112, −2.20810801527206915537135551564, −0.69981395771646563331595594602,
0.69981395771646563331595594602, 2.20810801527206915537135551564, 3.42783039718848135226359996112, 3.68408758135107676473088436769, 5.31048458901332118936358948508, 5.88758626226544082082133381731, 7.30541389423712184350038111377, 7.74336822653381720159826594241, 8.669076095804759980027435889706, 9.169312835666628577280917056470
