| L(s) = 1 | + 1.27·3-s − 4.07·5-s + 2.09·7-s − 1.38·9-s − 5.74·11-s + 4.45·13-s − 5.18·15-s + 1.29·17-s − 1.28·19-s + 2.66·21-s + 1.98·23-s + 11.6·25-s − 5.57·27-s − 0.687·29-s + 7.43·31-s − 7.30·33-s − 8.56·35-s − 9.14·37-s + 5.66·39-s + 6.98·41-s − 5.76·43-s + 5.65·45-s − 9.07·47-s − 2.59·49-s + 1.64·51-s − 11.0·53-s + 23.4·55-s + ⋯ |
| L(s) = 1 | + 0.733·3-s − 1.82·5-s + 0.793·7-s − 0.461·9-s − 1.73·11-s + 1.23·13-s − 1.33·15-s + 0.313·17-s − 0.294·19-s + 0.582·21-s + 0.413·23-s + 2.32·25-s − 1.07·27-s − 0.127·29-s + 1.33·31-s − 1.27·33-s − 1.44·35-s − 1.50·37-s + 0.906·39-s + 1.09·41-s − 0.879·43-s + 0.842·45-s − 1.32·47-s − 0.370·49-s + 0.229·51-s − 1.51·53-s + 3.15·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4016 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4016 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.393052488\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.393052488\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 251 | \( 1 - T \) |
| good | 3 | \( 1 - 1.27T + 3T^{2} \) |
| 5 | \( 1 + 4.07T + 5T^{2} \) |
| 7 | \( 1 - 2.09T + 7T^{2} \) |
| 11 | \( 1 + 5.74T + 11T^{2} \) |
| 13 | \( 1 - 4.45T + 13T^{2} \) |
| 17 | \( 1 - 1.29T + 17T^{2} \) |
| 19 | \( 1 + 1.28T + 19T^{2} \) |
| 23 | \( 1 - 1.98T + 23T^{2} \) |
| 29 | \( 1 + 0.687T + 29T^{2} \) |
| 31 | \( 1 - 7.43T + 31T^{2} \) |
| 37 | \( 1 + 9.14T + 37T^{2} \) |
| 41 | \( 1 - 6.98T + 41T^{2} \) |
| 43 | \( 1 + 5.76T + 43T^{2} \) |
| 47 | \( 1 + 9.07T + 47T^{2} \) |
| 53 | \( 1 + 11.0T + 53T^{2} \) |
| 59 | \( 1 - 13.3T + 59T^{2} \) |
| 61 | \( 1 - 0.329T + 61T^{2} \) |
| 67 | \( 1 - 8.50T + 67T^{2} \) |
| 71 | \( 1 - 8.72T + 71T^{2} \) |
| 73 | \( 1 - 15.8T + 73T^{2} \) |
| 79 | \( 1 + 6.44T + 79T^{2} \) |
| 83 | \( 1 - 13.9T + 83T^{2} \) |
| 89 | \( 1 + 3.38T + 89T^{2} \) |
| 97 | \( 1 - 5.52T + 97T^{2} \) |
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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
−8.373366807854520239434023870419, −7.970966657233607208961583847848, −7.35784558334963903583902157785, −6.34164558016984923141852714482, −5.16745309137062870344715955239, −4.69706221579348360417879215912, −3.54470592376005852495166939887, −3.25128730373064803063079179269, −2.14289195333873635960017017522, −0.63377295478491282391165570659,
0.63377295478491282391165570659, 2.14289195333873635960017017522, 3.25128730373064803063079179269, 3.54470592376005852495166939887, 4.69706221579348360417879215912, 5.16745309137062870344715955239, 6.34164558016984923141852714482, 7.35784558334963903583902157785, 7.970966657233607208961583847848, 8.373366807854520239434023870419
