| L(s) = 1 | + 2.27·2-s − 1.44·3-s + 3.16·4-s − 3.28·6-s − 7-s + 2.63·8-s − 0.906·9-s + 11-s − 4.57·12-s + 6.89·13-s − 2.27·14-s − 0.330·16-s − 3.21·17-s − 2.05·18-s + 4.16·19-s + 1.44·21-s + 2.27·22-s + 7.66·23-s − 3.81·24-s + 15.6·26-s + 5.65·27-s − 3.16·28-s + 1.86·29-s + 6.38·31-s − 6.02·32-s − 1.44·33-s − 7.30·34-s + ⋯ |
| L(s) = 1 | + 1.60·2-s − 0.835·3-s + 1.58·4-s − 1.34·6-s − 0.377·7-s + 0.932·8-s − 0.302·9-s + 0.301·11-s − 1.32·12-s + 1.91·13-s − 0.607·14-s − 0.0827·16-s − 0.779·17-s − 0.485·18-s + 0.954·19-s + 0.315·21-s + 0.484·22-s + 1.59·23-s − 0.778·24-s + 3.07·26-s + 1.08·27-s − 0.597·28-s + 0.346·29-s + 1.14·31-s − 1.06·32-s − 0.251·33-s − 1.25·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1925 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1925 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.348060394\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.348060394\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 \) |
| 7 | \( 1 + T \) |
| 11 | \( 1 - T \) |
| good | 2 | \( 1 - 2.27T + 2T^{2} \) |
| 3 | \( 1 + 1.44T + 3T^{2} \) |
| 13 | \( 1 - 6.89T + 13T^{2} \) |
| 17 | \( 1 + 3.21T + 17T^{2} \) |
| 19 | \( 1 - 4.16T + 19T^{2} \) |
| 23 | \( 1 - 7.66T + 23T^{2} \) |
| 29 | \( 1 - 1.86T + 29T^{2} \) |
| 31 | \( 1 - 6.38T + 31T^{2} \) |
| 37 | \( 1 + 6.43T + 37T^{2} \) |
| 41 | \( 1 - 10.3T + 41T^{2} \) |
| 43 | \( 1 + 5.53T + 43T^{2} \) |
| 47 | \( 1 + 4.01T + 47T^{2} \) |
| 53 | \( 1 - 11.7T + 53T^{2} \) |
| 59 | \( 1 - 10.3T + 59T^{2} \) |
| 61 | \( 1 - 11.4T + 61T^{2} \) |
| 67 | \( 1 + 10.9T + 67T^{2} \) |
| 71 | \( 1 + 11.7T + 71T^{2} \) |
| 73 | \( 1 + 9.90T + 73T^{2} \) |
| 79 | \( 1 - 14.8T + 79T^{2} \) |
| 83 | \( 1 + 5.07T + 83T^{2} \) |
| 89 | \( 1 + 0.508T + 89T^{2} \) |
| 97 | \( 1 - 4.91T + 97T^{2} \) |
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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
−9.065669672628466189904347949113, −8.504374104476564231130207758822, −7.01177951354532579486752494869, −6.53453787466379590132925939882, −5.83488578920733703951487449584, −5.25179478916784935279256493246, −4.35593058059418145506429977024, −3.49088518486916536458918639893, −2.73392388957624531328768465844, −1.06691331129411723155556127641,
1.06691331129411723155556127641, 2.73392388957624531328768465844, 3.49088518486916536458918639893, 4.35593058059418145506429977024, 5.25179478916784935279256493246, 5.83488578920733703951487449584, 6.53453787466379590132925939882, 7.01177951354532579486752494869, 8.504374104476564231130207758822, 9.065669672628466189904347949113
