| L(s) = 1 | + 1.96·2-s + 3-s + 1.85·4-s − 5-s + 1.96·6-s − 4.65·7-s − 0.286·8-s + 9-s − 1.96·10-s − 5.89·11-s + 1.85·12-s − 1.01·13-s − 9.13·14-s − 15-s − 4.27·16-s + 1.51·17-s + 1.96·18-s + 7.01·19-s − 1.85·20-s − 4.65·21-s − 11.5·22-s − 0.286·24-s + 25-s − 1.98·26-s + 27-s − 8.62·28-s − 0.247·29-s + ⋯ |
| L(s) = 1 | + 1.38·2-s + 0.577·3-s + 0.926·4-s − 0.447·5-s + 0.801·6-s − 1.75·7-s − 0.101·8-s + 0.333·9-s − 0.620·10-s − 1.77·11-s + 0.535·12-s − 0.281·13-s − 2.44·14-s − 0.258·15-s − 1.06·16-s + 0.366·17-s + 0.462·18-s + 1.60·19-s − 0.414·20-s − 1.01·21-s − 2.46·22-s − 0.0585·24-s + 0.200·25-s − 0.390·26-s + 0.192·27-s − 1.63·28-s − 0.0459·29-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 7935 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7935 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.834051547\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.834051547\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + T \) |
| 23 | \( 1 \) |
| good | 2 | \( 1 - 1.96T + 2T^{2} \) |
| 7 | \( 1 + 4.65T + 7T^{2} \) |
| 11 | \( 1 + 5.89T + 11T^{2} \) |
| 13 | \( 1 + 1.01T + 13T^{2} \) |
| 17 | \( 1 - 1.51T + 17T^{2} \) |
| 19 | \( 1 - 7.01T + 19T^{2} \) |
| 29 | \( 1 + 0.247T + 29T^{2} \) |
| 31 | \( 1 - 9.20T + 31T^{2} \) |
| 37 | \( 1 - 1.35T + 37T^{2} \) |
| 41 | \( 1 - 11.3T + 41T^{2} \) |
| 43 | \( 1 - 2.80T + 43T^{2} \) |
| 47 | \( 1 - 1.05T + 47T^{2} \) |
| 53 | \( 1 - 1.49T + 53T^{2} \) |
| 59 | \( 1 - 1.34T + 59T^{2} \) |
| 61 | \( 1 + 8.66T + 61T^{2} \) |
| 67 | \( 1 - 7.08T + 67T^{2} \) |
| 71 | \( 1 - 0.183T + 71T^{2} \) |
| 73 | \( 1 + 13.8T + 73T^{2} \) |
| 79 | \( 1 + 11.2T + 79T^{2} \) |
| 83 | \( 1 + 0.192T + 83T^{2} \) |
| 89 | \( 1 - 4.03T + 89T^{2} \) |
| 97 | \( 1 - 8.11T + 97T^{2} \) |
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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
−7.48281953360817008359583753692, −7.24824812581335778142962419013, −6.17952692044170395600489980885, −5.74273162779858269517945467705, −4.94724809065920859899252991426, −4.24239818162310153825853603918, −3.38122792376757694384885855285, −2.89144522293269415747915731086, −2.56400526244662169014367903169, −0.61532127903371023274550286285,
0.61532127903371023274550286285, 2.56400526244662169014367903169, 2.89144522293269415747915731086, 3.38122792376757694384885855285, 4.24239818162310153825853603918, 4.94724809065920859899252991426, 5.74273162779858269517945467705, 6.17952692044170395600489980885, 7.24824812581335778142962419013, 7.48281953360817008359583753692
