| L(s) = 1 | − 2.30·2-s − 3-s + 3.30·4-s + 2.30·6-s − 7-s − 3.00·8-s − 2·9-s − 1.60·11-s − 3.30·12-s + 2.30·14-s + 0.302·16-s − 7.60·17-s + 4.60·18-s + 5.60·19-s + 21-s + 3.69·22-s + 3·23-s + 3.00·24-s + 5·27-s − 3.30·28-s − 6.21·29-s + 4·31-s + 5.30·32-s + 1.60·33-s + 17.5·34-s − 6.60·36-s + 3.60·37-s + ⋯ |
| L(s) = 1 | − 1.62·2-s − 0.577·3-s + 1.65·4-s + 0.940·6-s − 0.377·7-s − 1.06·8-s − 0.666·9-s − 0.484·11-s − 0.953·12-s + 0.615·14-s + 0.0756·16-s − 1.84·17-s + 1.08·18-s + 1.28·19-s + 0.218·21-s + 0.788·22-s + 0.625·23-s + 0.612·24-s + 0.962·27-s − 0.624·28-s − 1.15·29-s + 0.718·31-s + 0.937·32-s + 0.279·33-s + 3.00·34-s − 1.10·36-s + 0.592·37-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4225 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4225 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 5 | \( 1 \) |
| 13 | \( 1 \) |
| good | 2 | \( 1 + 2.30T + 2T^{2} \) |
| 3 | \( 1 + T + 3T^{2} \) |
| 7 | \( 1 + T + 7T^{2} \) |
| 11 | \( 1 + 1.60T + 11T^{2} \) |
| 17 | \( 1 + 7.60T + 17T^{2} \) |
| 19 | \( 1 - 5.60T + 19T^{2} \) |
| 23 | \( 1 - 3T + 23T^{2} \) |
| 29 | \( 1 + 6.21T + 29T^{2} \) |
| 31 | \( 1 - 4T + 31T^{2} \) |
| 37 | \( 1 - 3.60T + 37T^{2} \) |
| 41 | \( 1 + 3T + 41T^{2} \) |
| 43 | \( 1 - 10.2T + 43T^{2} \) |
| 47 | \( 1 - 9.21T + 47T^{2} \) |
| 53 | \( 1 - 3.21T + 53T^{2} \) |
| 59 | \( 1 - 10.8T + 59T^{2} \) |
| 61 | \( 1 + T + 61T^{2} \) |
| 67 | \( 1 + 7T + 67T^{2} \) |
| 71 | \( 1 + 4.81T + 71T^{2} \) |
| 73 | \( 1 + 0.788T + 73T^{2} \) |
| 79 | \( 1 - 5.21T + 79T^{2} \) |
| 83 | \( 1 + 9.21T + 83T^{2} \) |
| 89 | \( 1 - 6.21T + 89T^{2} \) |
| 97 | \( 1 + 8.39T + 97T^{2} \) |
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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
−8.138974301942529904026999769557, −7.39452526759548797405133795775, −6.81812031924390525880533038960, −6.05317710539572665001307904888, −5.28840309853066241515613712858, −4.29404482096101849856557035076, −2.94158310033743530616353188925, −2.25334640849633247991995982842, −0.936578016626246354578999037112, 0,
0.936578016626246354578999037112, 2.25334640849633247991995982842, 2.94158310033743530616353188925, 4.29404482096101849856557035076, 5.28840309853066241515613712858, 6.05317710539572665001307904888, 6.81812031924390525880533038960, 7.39452526759548797405133795775, 8.138974301942529904026999769557
