| L(s) = 1 | + 2.35·2-s + 2.50·3-s + 3.56·4-s + 5.91·6-s + 7-s + 3.69·8-s + 3.28·9-s + 2.70·11-s + 8.94·12-s + 2.80·13-s + 2.35·14-s + 1.58·16-s − 1.99·17-s + 7.75·18-s − 1.11·19-s + 2.50·21-s + 6.37·22-s + 23-s + 9.26·24-s + 6.61·26-s + 0.721·27-s + 3.56·28-s − 8.42·29-s + 2.48·31-s − 3.65·32-s + 6.77·33-s − 4.71·34-s + ⋯ |
| L(s) = 1 | + 1.66·2-s + 1.44·3-s + 1.78·4-s + 2.41·6-s + 0.377·7-s + 1.30·8-s + 1.09·9-s + 0.814·11-s + 2.58·12-s + 0.777·13-s + 0.630·14-s + 0.395·16-s − 0.484·17-s + 1.82·18-s − 0.256·19-s + 0.547·21-s + 1.35·22-s + 0.208·23-s + 1.89·24-s + 1.29·26-s + 0.138·27-s + 0.673·28-s − 1.56·29-s + 0.445·31-s − 0.645·32-s + 1.17·33-s − 0.808·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4025 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4025 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(9.329211457\) |
| \(L(\frac12)\) |
\(\approx\) |
\(9.329211457\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 \) |
| 7 | \( 1 - T \) |
| 23 | \( 1 - T \) |
| good | 2 | \( 1 - 2.35T + 2T^{2} \) |
| 3 | \( 1 - 2.50T + 3T^{2} \) |
| 11 | \( 1 - 2.70T + 11T^{2} \) |
| 13 | \( 1 - 2.80T + 13T^{2} \) |
| 17 | \( 1 + 1.99T + 17T^{2} \) |
| 19 | \( 1 + 1.11T + 19T^{2} \) |
| 29 | \( 1 + 8.42T + 29T^{2} \) |
| 31 | \( 1 - 2.48T + 31T^{2} \) |
| 37 | \( 1 - 0.807T + 37T^{2} \) |
| 41 | \( 1 - 6.79T + 41T^{2} \) |
| 43 | \( 1 - 3.15T + 43T^{2} \) |
| 47 | \( 1 + 10.0T + 47T^{2} \) |
| 53 | \( 1 + 4.86T + 53T^{2} \) |
| 59 | \( 1 + 12.1T + 59T^{2} \) |
| 61 | \( 1 - 9.43T + 61T^{2} \) |
| 67 | \( 1 - 7.25T + 67T^{2} \) |
| 71 | \( 1 + 2.44T + 71T^{2} \) |
| 73 | \( 1 - 1.70T + 73T^{2} \) |
| 79 | \( 1 - 15.4T + 79T^{2} \) |
| 83 | \( 1 - 10.7T + 83T^{2} \) |
| 89 | \( 1 + 8.27T + 89T^{2} \) |
| 97 | \( 1 + 13.2T + 97T^{2} \) |
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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
−8.321839831976265034761899432257, −7.67511662941042836283243689260, −6.79709244783613653503741279800, −6.18717987775444684477750510573, −5.30488374659168381397844327432, −4.34581186673336239209520318965, −3.85723345767782802803569307626, −3.20999059500974890825177666838, −2.34170842106150334302819713500, −1.57596761518980431779946883151,
1.57596761518980431779946883151, 2.34170842106150334302819713500, 3.20999059500974890825177666838, 3.85723345767782802803569307626, 4.34581186673336239209520318965, 5.30488374659168381397844327432, 6.18717987775444684477750510573, 6.79709244783613653503741279800, 7.67511662941042836283243689260, 8.321839831976265034761899432257
