| L(s) = 1 | + 2.19·2-s − 3-s + 2.83·4-s − 2.19·6-s − 1.05·7-s + 1.82·8-s + 9-s − 3.22·11-s − 2.83·12-s − 2.58·13-s − 2.30·14-s − 1.64·16-s + 7.50·17-s + 2.19·18-s + 1.70·19-s + 1.05·21-s − 7.09·22-s + 2.21·23-s − 1.82·24-s − 5.69·26-s − 27-s − 2.97·28-s + 6.74·29-s + 10.8·31-s − 7.27·32-s + 3.22·33-s + 16.4·34-s + ⋯ |
| L(s) = 1 | + 1.55·2-s − 0.577·3-s + 1.41·4-s − 0.897·6-s − 0.397·7-s + 0.645·8-s + 0.333·9-s − 0.972·11-s − 0.817·12-s − 0.718·13-s − 0.617·14-s − 0.412·16-s + 1.81·17-s + 0.518·18-s + 0.390·19-s + 0.229·21-s − 1.51·22-s + 0.461·23-s − 0.372·24-s − 1.11·26-s − 0.192·27-s − 0.562·28-s + 1.25·29-s + 1.94·31-s − 1.28·32-s + 0.561·33-s + 2.82·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3525 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3525 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.441233251\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.441233251\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + T \) |
| 5 | \( 1 \) |
| 47 | \( 1 - T \) |
| good | 2 | \( 1 - 2.19T + 2T^{2} \) |
| 7 | \( 1 + 1.05T + 7T^{2} \) |
| 11 | \( 1 + 3.22T + 11T^{2} \) |
| 13 | \( 1 + 2.58T + 13T^{2} \) |
| 17 | \( 1 - 7.50T + 17T^{2} \) |
| 19 | \( 1 - 1.70T + 19T^{2} \) |
| 23 | \( 1 - 2.21T + 23T^{2} \) |
| 29 | \( 1 - 6.74T + 29T^{2} \) |
| 31 | \( 1 - 10.8T + 31T^{2} \) |
| 37 | \( 1 - 11.7T + 37T^{2} \) |
| 41 | \( 1 + 5.81T + 41T^{2} \) |
| 43 | \( 1 - 3.39T + 43T^{2} \) |
| 53 | \( 1 - 11.1T + 53T^{2} \) |
| 59 | \( 1 - 12.3T + 59T^{2} \) |
| 61 | \( 1 + 6.55T + 61T^{2} \) |
| 67 | \( 1 + 0.750T + 67T^{2} \) |
| 71 | \( 1 - 2.09T + 71T^{2} \) |
| 73 | \( 1 + 4.80T + 73T^{2} \) |
| 79 | \( 1 - 6.59T + 79T^{2} \) |
| 83 | \( 1 - 15.6T + 83T^{2} \) |
| 89 | \( 1 + 15.4T + 89T^{2} \) |
| 97 | \( 1 + 11.6T + 97T^{2} \) |
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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
−8.293782089453359006800887265718, −7.58735866033300743765769327926, −6.77849997126604260577696305712, −6.09077907060750993812594994003, −5.36979214995404915577406026432, −4.91736043353610390167996149103, −4.10066030795652079997908093119, −3.02699509680238855508779765168, −2.60544985784547051520649793336, −0.893617195737842274856141430542,
0.893617195737842274856141430542, 2.60544985784547051520649793336, 3.02699509680238855508779765168, 4.10066030795652079997908093119, 4.91736043353610390167996149103, 5.36979214995404915577406026432, 6.09077907060750993812594994003, 6.77849997126604260577696305712, 7.58735866033300743765769327926, 8.293782089453359006800887265718
