| L(s) = 1 | + 1.21·2-s − 0.525·4-s − 2.59·7-s − 3.06·8-s + 6.11·11-s − 13-s − 3.14·14-s − 2.67·16-s − 4.37·17-s + 4.14·19-s + 7.42·22-s − 7.95·23-s − 1.21·26-s + 1.36·28-s + 3·29-s + 5.36·31-s + 2.88·32-s − 5.31·34-s + 6.90·37-s + 5.03·38-s + 9.19·41-s + 11.1·43-s − 3.21·44-s − 9.65·46-s + 1.21·47-s − 0.280·49-s + 0.525·52-s + ⋯ |
| L(s) = 1 | + 0.858·2-s − 0.262·4-s − 0.979·7-s − 1.08·8-s + 1.84·11-s − 0.277·13-s − 0.841·14-s − 0.668·16-s − 1.06·17-s + 0.951·19-s + 1.58·22-s − 1.65·23-s − 0.238·26-s + 0.257·28-s + 0.557·29-s + 0.963·31-s + 0.510·32-s − 0.911·34-s + 1.13·37-s + 0.817·38-s + 1.43·41-s + 1.69·43-s − 0.484·44-s − 1.42·46-s + 0.177·47-s − 0.0401·49-s + 0.0728·52-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2925 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2925 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.091331447\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.091331447\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 5 | \( 1 \) |
| 13 | \( 1 + T \) |
| good | 2 | \( 1 - 1.21T + 2T^{2} \) |
| 7 | \( 1 + 2.59T + 7T^{2} \) |
| 11 | \( 1 - 6.11T + 11T^{2} \) |
| 17 | \( 1 + 4.37T + 17T^{2} \) |
| 19 | \( 1 - 4.14T + 19T^{2} \) |
| 23 | \( 1 + 7.95T + 23T^{2} \) |
| 29 | \( 1 - 3T + 29T^{2} \) |
| 31 | \( 1 - 5.36T + 31T^{2} \) |
| 37 | \( 1 - 6.90T + 37T^{2} \) |
| 41 | \( 1 - 9.19T + 41T^{2} \) |
| 43 | \( 1 - 11.1T + 43T^{2} \) |
| 47 | \( 1 - 1.21T + 47T^{2} \) |
| 53 | \( 1 + 4.95T + 53T^{2} \) |
| 59 | \( 1 + 5.44T + 59T^{2} \) |
| 61 | \( 1 - 9.99T + 61T^{2} \) |
| 67 | \( 1 + 4.87T + 67T^{2} \) |
| 71 | \( 1 + 1.39T + 71T^{2} \) |
| 73 | \( 1 - 13.1T + 73T^{2} \) |
| 79 | \( 1 + 12.3T + 79T^{2} \) |
| 83 | \( 1 - 13.2T + 83T^{2} \) |
| 89 | \( 1 + 8.04T + 89T^{2} \) |
| 97 | \( 1 - 15.3T + 97T^{2} \) |
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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
−8.991751318986470655973280385267, −8.013368954892532267285102216790, −6.95192680120421601993874948760, −6.18442075753557920256576509372, −5.94065758359578828086230637967, −4.55507049256957457691650705857, −4.14686231331310590194557677356, −3.35052482293910487789662006925, −2.39075488082651832022698515384, −0.77943574852935345861149095536,
0.77943574852935345861149095536, 2.39075488082651832022698515384, 3.35052482293910487789662006925, 4.14686231331310590194557677356, 4.55507049256957457691650705857, 5.94065758359578828086230637967, 6.18442075753557920256576509372, 6.95192680120421601993874948760, 8.013368954892532267285102216790, 8.991751318986470655973280385267
