| L(s) = 1 | − 4.88·2-s + 3·3-s + 15.8·4-s − 14.6·6-s − 5.48·7-s − 38.3·8-s + 9·9-s + 50.1·11-s + 47.5·12-s + 20.7·13-s + 26.7·14-s + 60.3·16-s + 18.8·17-s − 43.9·18-s + 78.8·19-s − 16.4·21-s − 244.·22-s + 6.33·23-s − 114.·24-s − 101.·26-s + 27·27-s − 86.9·28-s + 29·29-s + 310.·31-s + 11.9·32-s + 150.·33-s − 91.8·34-s + ⋯ |
| L(s) = 1 | − 1.72·2-s + 0.577·3-s + 1.98·4-s − 0.996·6-s − 0.296·7-s − 1.69·8-s + 0.333·9-s + 1.37·11-s + 1.14·12-s + 0.442·13-s + 0.511·14-s + 0.942·16-s + 0.268·17-s − 0.575·18-s + 0.951·19-s − 0.171·21-s − 2.37·22-s + 0.0574·23-s − 0.977·24-s − 0.763·26-s + 0.192·27-s − 0.586·28-s + 0.185·29-s + 1.80·31-s + 0.0658·32-s + 0.793·33-s − 0.463·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2175 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2175 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(1.551133730\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.551133730\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 - 3T \) |
| 5 | \( 1 \) |
| 29 | \( 1 - 29T \) |
| good | 2 | \( 1 + 4.88T + 8T^{2} \) |
| 7 | \( 1 + 5.48T + 343T^{2} \) |
| 11 | \( 1 - 50.1T + 1.33e3T^{2} \) |
| 13 | \( 1 - 20.7T + 2.19e3T^{2} \) |
| 17 | \( 1 - 18.8T + 4.91e3T^{2} \) |
| 19 | \( 1 - 78.8T + 6.85e3T^{2} \) |
| 23 | \( 1 - 6.33T + 1.21e4T^{2} \) |
| 31 | \( 1 - 310.T + 2.97e4T^{2} \) |
| 37 | \( 1 - 338.T + 5.06e4T^{2} \) |
| 41 | \( 1 - 353.T + 6.89e4T^{2} \) |
| 43 | \( 1 + 507.T + 7.95e4T^{2} \) |
| 47 | \( 1 - 112.T + 1.03e5T^{2} \) |
| 53 | \( 1 - 144.T + 1.48e5T^{2} \) |
| 59 | \( 1 + 342.T + 2.05e5T^{2} \) |
| 61 | \( 1 - 357.T + 2.26e5T^{2} \) |
| 67 | \( 1 - 183.T + 3.00e5T^{2} \) |
| 71 | \( 1 + 594.T + 3.57e5T^{2} \) |
| 73 | \( 1 - 622.T + 3.89e5T^{2} \) |
| 79 | \( 1 + 1.27e3T + 4.93e5T^{2} \) |
| 83 | \( 1 - 739.T + 5.71e5T^{2} \) |
| 89 | \( 1 - 906.T + 7.04e5T^{2} \) |
| 97 | \( 1 + 76.0T + 9.12e5T^{2} \) |
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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
−8.740702952409593441791730488902, −8.155155507745196981347340952664, −7.44666341865962883798538255354, −6.63167190322996560624689987003, −6.08732128491885148959443709137, −4.55329964158450434411195405496, −3.45718853802695024444683386522, −2.57461052069475878111473428581, −1.40210898356231416426537418545, −0.806841764287716520325264934891,
0.806841764287716520325264934891, 1.40210898356231416426537418545, 2.57461052069475878111473428581, 3.45718853802695024444683386522, 4.55329964158450434411195405496, 6.08732128491885148959443709137, 6.63167190322996560624689987003, 7.44666341865962883798538255354, 8.155155507745196981347340952664, 8.740702952409593441791730488902
