| L(s) = 1 | + 1.90·2-s − 0.618·3-s + 2.61·4-s − 1.17·6-s + 7-s + 3.07·8-s − 0.618·9-s − 1.61·12-s + 1.90·14-s + 3.23·16-s − 17-s − 1.17·18-s − 0.618·21-s − 1.90·24-s + 27-s + 2.61·28-s + 1.17·31-s + 3.07·32-s − 1.90·34-s − 1.61·36-s − 1.90·41-s − 1.17·42-s − 1.90·43-s − 1.99·48-s + 49-s + 0.618·51-s − 1.17·53-s + ⋯ |
| L(s) = 1 | + 1.90·2-s − 0.618·3-s + 2.61·4-s − 1.17·6-s + 7-s + 3.07·8-s − 0.618·9-s − 1.61·12-s + 1.90·14-s + 3.23·16-s − 17-s − 1.17·18-s − 0.618·21-s − 1.90·24-s + 27-s + 2.61·28-s + 1.17·31-s + 3.07·32-s − 1.90·34-s − 1.61·36-s − 1.90·41-s − 1.17·42-s − 1.90·43-s − 1.99·48-s + 49-s + 0.618·51-s − 1.17·53-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2975 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2975 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{1}{2})\) |
\(\approx\) |
\(3.457606278\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.457606278\) |
| \(L(1)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 \) |
| 7 | \( 1 - T \) |
| 17 | \( 1 + T \) |
| good | 2 | \( 1 - 1.90T + T^{2} \) |
| 3 | \( 1 + 0.618T + T^{2} \) |
| 11 | \( 1 - T^{2} \) |
| 13 | \( 1 - T^{2} \) |
| 19 | \( 1 - T^{2} \) |
| 23 | \( 1 - T^{2} \) |
| 29 | \( 1 - T^{2} \) |
| 31 | \( 1 - 1.17T + T^{2} \) |
| 37 | \( 1 - T^{2} \) |
| 41 | \( 1 + 1.90T + T^{2} \) |
| 43 | \( 1 + 1.90T + T^{2} \) |
| 47 | \( 1 - T^{2} \) |
| 53 | \( 1 + 1.17T + T^{2} \) |
| 59 | \( 1 - T^{2} \) |
| 61 | \( 1 - 1.90T + T^{2} \) |
| 67 | \( 1 + 1.17T + T^{2} \) |
| 71 | \( 1 - T^{2} \) |
| 73 | \( 1 - 0.618T + T^{2} \) |
| 79 | \( 1 - T^{2} \) |
| 83 | \( 1 - T^{2} \) |
| 89 | \( 1 - T^{2} \) |
| 97 | \( 1 + 1.61T + T^{2} \) |
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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
−8.606316252552294589773983717405, −8.019402210859221312739993906650, −6.88275980335931231896922065842, −6.49217414045292756132050353970, −5.60631845554387230220255370788, −4.97597473751703061477659953570, −4.54101629127788644027699811582, −3.50861380214116997618827502789, −2.58682823608431528421569254676, −1.63729988211476606281415168571,
1.63729988211476606281415168571, 2.58682823608431528421569254676, 3.50861380214116997618827502789, 4.54101629127788644027699811582, 4.97597473751703061477659953570, 5.60631845554387230220255370788, 6.49217414045292756132050353970, 6.88275980335931231896922065842, 8.019402210859221312739993906650, 8.606316252552294589773983717405
