| L(s) = 1 | − 2.24·2-s − 3-s + 3.04·4-s + 2.24·6-s + 7-s − 2.35·8-s + 9-s + 1.24·11-s − 3.04·12-s − 2.24·14-s − 0.801·16-s + 0.890·17-s − 2.24·18-s + 4.71·19-s − 21-s − 2.80·22-s − 0.225·23-s + 2.35·24-s − 5·25-s − 27-s + 3.04·28-s − 7.40·29-s + 0.396·31-s + 6.51·32-s − 1.24·33-s − 2·34-s + 3.04·36-s + ⋯ |
| L(s) = 1 | − 1.58·2-s − 0.577·3-s + 1.52·4-s + 0.917·6-s + 0.377·7-s − 0.833·8-s + 0.333·9-s + 0.375·11-s − 0.880·12-s − 0.600·14-s − 0.200·16-s + 0.215·17-s − 0.529·18-s + 1.08·19-s − 0.218·21-s − 0.597·22-s − 0.0469·23-s + 0.481·24-s − 25-s − 0.192·27-s + 0.576·28-s − 1.37·29-s + 0.0711·31-s + 1.15·32-s − 0.217·33-s − 0.342·34-s + 0.508·36-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3549 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3549 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + T \) |
| 7 | \( 1 - T \) |
| 13 | \( 1 \) |
| good | 2 | \( 1 + 2.24T + 2T^{2} \) |
| 5 | \( 1 + 5T^{2} \) |
| 11 | \( 1 - 1.24T + 11T^{2} \) |
| 17 | \( 1 - 0.890T + 17T^{2} \) |
| 19 | \( 1 - 4.71T + 19T^{2} \) |
| 23 | \( 1 + 0.225T + 23T^{2} \) |
| 29 | \( 1 + 7.40T + 29T^{2} \) |
| 31 | \( 1 - 0.396T + 31T^{2} \) |
| 37 | \( 1 - 3.65T + 37T^{2} \) |
| 41 | \( 1 + 3.50T + 41T^{2} \) |
| 43 | \( 1 - 3.08T + 43T^{2} \) |
| 47 | \( 1 + 8.81T + 47T^{2} \) |
| 53 | \( 1 + 10.3T + 53T^{2} \) |
| 59 | \( 1 + 7.20T + 59T^{2} \) |
| 61 | \( 1 + 8.37T + 61T^{2} \) |
| 67 | \( 1 - 9.15T + 67T^{2} \) |
| 71 | \( 1 + 2.44T + 71T^{2} \) |
| 73 | \( 1 + 6.93T + 73T^{2} \) |
| 79 | \( 1 - 5.12T + 79T^{2} \) |
| 83 | \( 1 - 8.59T + 83T^{2} \) |
| 89 | \( 1 - 0.670T + 89T^{2} \) |
| 97 | \( 1 - 8.31T + 97T^{2} \) |
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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
−7.949786454235079615998526346311, −7.82464450422104264392474080330, −6.93318401538512404147317053965, −6.18079759004290643897250482725, −5.34883723814263430799448936418, −4.40292752971199819964134375934, −3.28756720488356355281657739524, −1.95876004728810180744956589043, −1.22626367335038316231087646450, 0,
1.22626367335038316231087646450, 1.95876004728810180744956589043, 3.28756720488356355281657739524, 4.40292752971199819964134375934, 5.34883723814263430799448936418, 6.18079759004290643897250482725, 6.93318401538512404147317053965, 7.82464450422104264392474080330, 7.949786454235079615998526346311
