| L(s) = 1 | + 2-s + 3.21·3-s + 4-s + 3.21·6-s + 7-s + 8-s + 7.34·9-s + 2.38·11-s + 3.21·12-s + 4.44·13-s + 14-s + 16-s − 17-s + 7.34·18-s + 1.27·19-s + 3.21·21-s + 2.38·22-s − 5.42·23-s + 3.21·24-s + 4.44·26-s + 13.9·27-s + 28-s + 4.02·29-s − 1.53·31-s + 32-s + 7.66·33-s − 34-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 1.85·3-s + 0.5·4-s + 1.31·6-s + 0.377·7-s + 0.353·8-s + 2.44·9-s + 0.718·11-s + 0.928·12-s + 1.23·13-s + 0.267·14-s + 0.250·16-s − 0.242·17-s + 1.73·18-s + 0.293·19-s + 0.701·21-s + 0.507·22-s − 1.13·23-s + 0.656·24-s + 0.872·26-s + 2.69·27-s + 0.188·28-s + 0.747·29-s − 0.275·31-s + 0.176·32-s + 1.33·33-s − 0.171·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5950 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5950 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(7.643928561\) |
| \(L(\frac12)\) |
\(\approx\) |
\(7.643928561\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - T \) |
| 5 | \( 1 \) |
| 7 | \( 1 - T \) |
| 17 | \( 1 + T \) |
| good | 3 | \( 1 - 3.21T + 3T^{2} \) |
| 11 | \( 1 - 2.38T + 11T^{2} \) |
| 13 | \( 1 - 4.44T + 13T^{2} \) |
| 19 | \( 1 - 1.27T + 19T^{2} \) |
| 23 | \( 1 + 5.42T + 23T^{2} \) |
| 29 | \( 1 - 4.02T + 29T^{2} \) |
| 31 | \( 1 + 1.53T + 31T^{2} \) |
| 37 | \( 1 + 8.76T + 37T^{2} \) |
| 41 | \( 1 + 6.98T + 41T^{2} \) |
| 43 | \( 1 + 6.32T + 43T^{2} \) |
| 47 | \( 1 + 1.72T + 47T^{2} \) |
| 53 | \( 1 - 0.864T + 53T^{2} \) |
| 59 | \( 1 + 10.1T + 59T^{2} \) |
| 61 | \( 1 + 4.54T + 61T^{2} \) |
| 67 | \( 1 + 10.3T + 67T^{2} \) |
| 71 | \( 1 + 6.09T + 71T^{2} \) |
| 73 | \( 1 + 8.79T + 73T^{2} \) |
| 79 | \( 1 - 11.5T + 79T^{2} \) |
| 83 | \( 1 + 4.16T + 83T^{2} \) |
| 89 | \( 1 - 13.3T + 89T^{2} \) |
| 97 | \( 1 + 3.78T + 97T^{2} \) |
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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
−8.130360575080127066811877218980, −7.49854016572142663452824488066, −6.71772298634098410258815391944, −6.08825767387408907127144771085, −4.91104850590563196275442097855, −4.19956464850375196731718701609, −3.53168209927624763997847462540, −3.07597764243986878182515797353, −1.90558947881845588501168325862, −1.47521078590216649651823486118,
1.47521078590216649651823486118, 1.90558947881845588501168325862, 3.07597764243986878182515797353, 3.53168209927624763997847462540, 4.19956464850375196731718701609, 4.91104850590563196275442097855, 6.08825767387408907127144771085, 6.71772298634098410258815391944, 7.49854016572142663452824488066, 8.130360575080127066811877218980
