| L(s) = 1 | − 2-s − 0.0752·3-s + 4-s − 2.78·5-s + 0.0752·6-s − 7-s − 8-s − 2.99·9-s + 2.78·10-s + 2.08·11-s − 0.0752·12-s − 3.50·13-s + 14-s + 0.209·15-s + 16-s − 5.12·17-s + 2.99·18-s + 1.33·19-s − 2.78·20-s + 0.0752·21-s − 2.08·22-s − 0.969·23-s + 0.0752·24-s + 2.77·25-s + 3.50·26-s + 0.450·27-s − 28-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 0.0434·3-s + 0.5·4-s − 1.24·5-s + 0.0307·6-s − 0.377·7-s − 0.353·8-s − 0.998·9-s + 0.881·10-s + 0.630·11-s − 0.0217·12-s − 0.971·13-s + 0.267·14-s + 0.0541·15-s + 0.250·16-s − 1.24·17-s + 0.705·18-s + 0.305·19-s − 0.623·20-s + 0.0164·21-s − 0.445·22-s − 0.202·23-s + 0.0153·24-s + 0.555·25-s + 0.686·26-s + 0.0867·27-s − 0.188·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6034 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6034 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.1722820628\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.1722820628\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + T \) |
| 7 | \( 1 + T \) |
| 431 | \( 1 - T \) |
| good | 3 | \( 1 + 0.0752T + 3T^{2} \) |
| 5 | \( 1 + 2.78T + 5T^{2} \) |
| 11 | \( 1 - 2.08T + 11T^{2} \) |
| 13 | \( 1 + 3.50T + 13T^{2} \) |
| 17 | \( 1 + 5.12T + 17T^{2} \) |
| 19 | \( 1 - 1.33T + 19T^{2} \) |
| 23 | \( 1 + 0.969T + 23T^{2} \) |
| 29 | \( 1 + 0.965T + 29T^{2} \) |
| 31 | \( 1 - 2.05T + 31T^{2} \) |
| 37 | \( 1 + 4.89T + 37T^{2} \) |
| 41 | \( 1 + 7.97T + 41T^{2} \) |
| 43 | \( 1 + 11.0T + 43T^{2} \) |
| 47 | \( 1 + 0.588T + 47T^{2} \) |
| 53 | \( 1 + 0.927T + 53T^{2} \) |
| 59 | \( 1 - 4.91T + 59T^{2} \) |
| 61 | \( 1 + 9.55T + 61T^{2} \) |
| 67 | \( 1 + 15.9T + 67T^{2} \) |
| 71 | \( 1 + 6.13T + 71T^{2} \) |
| 73 | \( 1 - 11.5T + 73T^{2} \) |
| 79 | \( 1 + 10.3T + 79T^{2} \) |
| 83 | \( 1 + 12.9T + 83T^{2} \) |
| 89 | \( 1 - 12.4T + 89T^{2} \) |
| 97 | \( 1 + 7.10T + 97T^{2} \) |
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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
−8.197113425073264853068418773060, −7.40265712292987040324065043940, −6.84606068427823887238252193942, −6.19877781755302044246505861322, −5.17379105374688434046517474545, −4.39211894490977791433795562643, −3.47850262425585156520147933483, −2.84187740451512601593693608175, −1.75731492826400539770316054068, −0.23172483262533671477432803943,
0.23172483262533671477432803943, 1.75731492826400539770316054068, 2.84187740451512601593693608175, 3.47850262425585156520147933483, 4.39211894490977791433795562643, 5.17379105374688434046517474545, 6.19877781755302044246505861322, 6.84606068427823887238252193942, 7.40265712292987040324065043940, 8.197113425073264853068418773060
