| L(s) = 1 | + 3-s − 2·4-s − 5-s − 7-s + 9-s − 3·11-s − 2·12-s + 13-s − 15-s + 4·16-s − 3·17-s + 2·19-s + 2·20-s − 21-s + 3·23-s + 25-s + 27-s + 2·28-s + 31-s − 3·33-s + 35-s − 2·36-s − 7·37-s + 39-s − 9·41-s − 10·43-s + 6·44-s + ⋯ |
| L(s) = 1 | + 0.577·3-s − 4-s − 0.447·5-s − 0.377·7-s + 1/3·9-s − 0.904·11-s − 0.577·12-s + 0.277·13-s − 0.258·15-s + 16-s − 0.727·17-s + 0.458·19-s + 0.447·20-s − 0.218·21-s + 0.625·23-s + 1/5·25-s + 0.192·27-s + 0.377·28-s + 0.179·31-s − 0.522·33-s + 0.169·35-s − 1/3·36-s − 1.15·37-s + 0.160·39-s − 1.40·41-s − 1.52·43-s + 0.904·44-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6045 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6045 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.134212625\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.134212625\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 - T \) |
| 5 | \( 1 + T \) |
| 13 | \( 1 - T \) |
| 31 | \( 1 - T \) |
| good | 2 | \( 1 + p T^{2} \) |
| 7 | \( 1 + T + p T^{2} \) |
| 11 | \( 1 + 3 T + p T^{2} \) |
| 17 | \( 1 + 3 T + p T^{2} \) |
| 19 | \( 1 - 2 T + p T^{2} \) |
| 23 | \( 1 - 3 T + p T^{2} \) |
| 29 | \( 1 + p T^{2} \) |
| 37 | \( 1 + 7 T + p T^{2} \) |
| 41 | \( 1 + 9 T + p T^{2} \) |
| 43 | \( 1 + 10 T + p T^{2} \) |
| 47 | \( 1 - 12 T + p T^{2} \) |
| 53 | \( 1 + 3 T + p T^{2} \) |
| 59 | \( 1 - 12 T + p T^{2} \) |
| 61 | \( 1 + T + p T^{2} \) |
| 67 | \( 1 + 4 T + p T^{2} \) |
| 71 | \( 1 + 9 T + p T^{2} \) |
| 73 | \( 1 - 2 T + p T^{2} \) |
| 79 | \( 1 + T + p T^{2} \) |
| 83 | \( 1 + 18 T + p T^{2} \) |
| 89 | \( 1 - 15 T + p T^{2} \) |
| 97 | \( 1 - 17 T + p T^{2} \) |
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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
−8.329505882528099456631445275243, −7.42903980837140067522789440470, −6.87245909228288937763530243911, −5.81863293005061305975785369247, −5.01415283154477736879111848822, −4.48345379476440159851870151791, −3.48633135418296489890274782867, −3.10094354832761761562446618161, −1.86463862240861561376209124835, −0.53811607694792309300370208095,
0.53811607694792309300370208095, 1.86463862240861561376209124835, 3.10094354832761761562446618161, 3.48633135418296489890274782867, 4.48345379476440159851870151791, 5.01415283154477736879111848822, 5.81863293005061305975785369247, 6.87245909228288937763530243911, 7.42903980837140067522789440470, 8.329505882528099456631445275243
