| L(s) = 1 | + 1.89·2-s − 2.61·3-s + 1.57·4-s − 4.93·6-s − 0.975·7-s − 0.804·8-s + 3.81·9-s + 2.71·11-s − 4.11·12-s + 4.45·13-s − 1.84·14-s − 4.67·16-s − 3.87·17-s + 7.21·18-s + 2.54·21-s + 5.14·22-s + 7.30·23-s + 2.10·24-s + 8.41·26-s − 2.13·27-s − 1.53·28-s − 4.78·29-s + 0.708·31-s − 7.21·32-s − 7.10·33-s − 7.32·34-s + 6.01·36-s + ⋯ |
| L(s) = 1 | + 1.33·2-s − 1.50·3-s + 0.787·4-s − 2.01·6-s − 0.368·7-s − 0.284·8-s + 1.27·9-s + 0.819·11-s − 1.18·12-s + 1.23·13-s − 0.492·14-s − 1.16·16-s − 0.939·17-s + 1.70·18-s + 0.555·21-s + 1.09·22-s + 1.52·23-s + 0.428·24-s + 1.65·26-s − 0.411·27-s − 0.290·28-s − 0.889·29-s + 0.127·31-s − 1.27·32-s − 1.23·33-s − 1.25·34-s + 1.00·36-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 9025 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 9025 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.070006540\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.070006540\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 \) |
| 19 | \( 1 \) |
| good | 2 | \( 1 - 1.89T + 2T^{2} \) |
| 3 | \( 1 + 2.61T + 3T^{2} \) |
| 7 | \( 1 + 0.975T + 7T^{2} \) |
| 11 | \( 1 - 2.71T + 11T^{2} \) |
| 13 | \( 1 - 4.45T + 13T^{2} \) |
| 17 | \( 1 + 3.87T + 17T^{2} \) |
| 23 | \( 1 - 7.30T + 23T^{2} \) |
| 29 | \( 1 + 4.78T + 29T^{2} \) |
| 31 | \( 1 - 0.708T + 31T^{2} \) |
| 37 | \( 1 - 11.8T + 37T^{2} \) |
| 41 | \( 1 + 0.268T + 41T^{2} \) |
| 43 | \( 1 + 11.1T + 43T^{2} \) |
| 47 | \( 1 - 4.36T + 47T^{2} \) |
| 53 | \( 1 + 6.83T + 53T^{2} \) |
| 59 | \( 1 + 6.89T + 59T^{2} \) |
| 61 | \( 1 + 11.0T + 61T^{2} \) |
| 67 | \( 1 - 0.770T + 67T^{2} \) |
| 71 | \( 1 - 6.30T + 71T^{2} \) |
| 73 | \( 1 - 13.3T + 73T^{2} \) |
| 79 | \( 1 + 7.47T + 79T^{2} \) |
| 83 | \( 1 + 7.30T + 83T^{2} \) |
| 89 | \( 1 - 1.48T + 89T^{2} \) |
| 97 | \( 1 - 6.41T + 97T^{2} \) |
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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
−7.21953809453983363268497188156, −6.57756677747604622941723059449, −6.19392232339878647890628828939, −5.77473613660442538246171593616, −4.82144622132466324340872387984, −4.54514916790546534215878394635, −3.65787211478287238768630754604, −3.02540209876023755905773040102, −1.69785473512376769623280891039, −0.62331021738716217939270960354,
0.62331021738716217939270960354, 1.69785473512376769623280891039, 3.02540209876023755905773040102, 3.65787211478287238768630754604, 4.54514916790546534215878394635, 4.82144622132466324340872387984, 5.77473613660442538246171593616, 6.19392232339878647890628828939, 6.57756677747604622941723059449, 7.21953809453983363268497188156
