| L(s) = 1 | + 2-s + 4-s + 0.521·5-s − 3.80·7-s + 8-s + 0.521·10-s − 2.59·11-s − 5.94·13-s − 3.80·14-s + 16-s − 4.15·17-s + 5.07·19-s + 0.521·20-s − 2.59·22-s − 3.81·23-s − 4.72·25-s − 5.94·26-s − 3.80·28-s + 3.56·29-s − 0.333·31-s + 32-s − 4.15·34-s − 1.98·35-s + 10.9·37-s + 5.07·38-s + 0.521·40-s + 8.72·41-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 0.5·4-s + 0.233·5-s − 1.43·7-s + 0.353·8-s + 0.164·10-s − 0.782·11-s − 1.64·13-s − 1.01·14-s + 0.250·16-s − 1.00·17-s + 1.16·19-s + 0.116·20-s − 0.553·22-s − 0.794·23-s − 0.945·25-s − 1.16·26-s − 0.718·28-s + 0.661·29-s − 0.0599·31-s + 0.176·32-s − 0.712·34-s − 0.335·35-s + 1.80·37-s + 0.823·38-s + 0.0823·40-s + 1.36·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8046 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8046 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.837654625\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.837654625\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - T \) |
| 3 | \( 1 \) |
| 149 | \( 1 + T \) |
| good | 5 | \( 1 - 0.521T + 5T^{2} \) |
| 7 | \( 1 + 3.80T + 7T^{2} \) |
| 11 | \( 1 + 2.59T + 11T^{2} \) |
| 13 | \( 1 + 5.94T + 13T^{2} \) |
| 17 | \( 1 + 4.15T + 17T^{2} \) |
| 19 | \( 1 - 5.07T + 19T^{2} \) |
| 23 | \( 1 + 3.81T + 23T^{2} \) |
| 29 | \( 1 - 3.56T + 29T^{2} \) |
| 31 | \( 1 + 0.333T + 31T^{2} \) |
| 37 | \( 1 - 10.9T + 37T^{2} \) |
| 41 | \( 1 - 8.72T + 41T^{2} \) |
| 43 | \( 1 + 2.32T + 43T^{2} \) |
| 47 | \( 1 - 4.62T + 47T^{2} \) |
| 53 | \( 1 + 6.24T + 53T^{2} \) |
| 59 | \( 1 - 14.2T + 59T^{2} \) |
| 61 | \( 1 - 9.85T + 61T^{2} \) |
| 67 | \( 1 - 0.999T + 67T^{2} \) |
| 71 | \( 1 - 2.76T + 71T^{2} \) |
| 73 | \( 1 + 1.75T + 73T^{2} \) |
| 79 | \( 1 - 1.63T + 79T^{2} \) |
| 83 | \( 1 + 15.4T + 83T^{2} \) |
| 89 | \( 1 - 4.25T + 89T^{2} \) |
| 97 | \( 1 - 7.53T + 97T^{2} \) |
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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
−7.48417198708676815454176021294, −7.19211079881849022905504209935, −6.25522125314270900998080410214, −5.83401673855133905701222519676, −5.01712245537113337434790078118, −4.33045030702975139116940351345, −3.48795445974005220324712290665, −2.57165666922034456255224837637, −2.31847701790704439968011748912, −0.55768597622482845535817162715,
0.55768597622482845535817162715, 2.31847701790704439968011748912, 2.57165666922034456255224837637, 3.48795445974005220324712290665, 4.33045030702975139116940351345, 5.01712245537113337434790078118, 5.83401673855133905701222519676, 6.25522125314270900998080410214, 7.19211079881849022905504209935, 7.48417198708676815454176021294
