| L(s) = 1 | − 2·2-s + 3·3-s + 4·4-s + 5·5-s − 6·6-s − 8·8-s + 9·9-s − 10·10-s − 32·11-s + 12·12-s + 15·13-s + 15·15-s + 16·16-s − 70·17-s − 18·18-s + 15·19-s + 20·20-s + 64·22-s − 42·23-s − 24·24-s + 25·25-s − 30·26-s + 27·27-s + 90·29-s − 30·30-s − 85·31-s − 32·32-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 0.577·3-s + 1/2·4-s + 0.447·5-s − 0.408·6-s − 0.353·8-s + 1/3·9-s − 0.316·10-s − 0.877·11-s + 0.288·12-s + 0.320·13-s + 0.258·15-s + 1/4·16-s − 0.998·17-s − 0.235·18-s + 0.181·19-s + 0.223·20-s + 0.620·22-s − 0.380·23-s − 0.204·24-s + 1/5·25-s − 0.226·26-s + 0.192·27-s + 0.576·29-s − 0.182·30-s − 0.492·31-s − 0.176·32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1470 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1470 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(1.876138808\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.876138808\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + p T \) |
| 3 | \( 1 - p T \) |
| 5 | \( 1 - p T \) |
| 7 | \( 1 \) |
| good | 11 | \( 1 + 32 T + p^{3} T^{2} \) |
| 13 | \( 1 - 15 T + p^{3} T^{2} \) |
| 17 | \( 1 + 70 T + p^{3} T^{2} \) |
| 19 | \( 1 - 15 T + p^{3} T^{2} \) |
| 23 | \( 1 + 42 T + p^{3} T^{2} \) |
| 29 | \( 1 - 90 T + p^{3} T^{2} \) |
| 31 | \( 1 + 85 T + p^{3} T^{2} \) |
| 37 | \( 1 - 113 T + p^{3} T^{2} \) |
| 41 | \( 1 - 4 p T + p^{3} T^{2} \) |
| 43 | \( 1 - 169 T + p^{3} T^{2} \) |
| 47 | \( 1 - 326 T + p^{3} T^{2} \) |
| 53 | \( 1 + 44 T + p^{3} T^{2} \) |
| 59 | \( 1 + 782 T + p^{3} T^{2} \) |
| 61 | \( 1 - 658 T + p^{3} T^{2} \) |
| 67 | \( 1 - 1071 T + p^{3} T^{2} \) |
| 71 | \( 1 - 344 T + p^{3} T^{2} \) |
| 73 | \( 1 - 431 T + p^{3} T^{2} \) |
| 79 | \( 1 - 397 T + p^{3} T^{2} \) |
| 83 | \( 1 - 680 T + p^{3} T^{2} \) |
| 89 | \( 1 - 1534 T + p^{3} T^{2} \) |
| 97 | \( 1 + 1234 T + p^{3} T^{2} \) |
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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
−9.188362947064529927947140817078, −8.359659388074704985756943902059, −7.74949124025404614478212183160, −6.85368690328549153891306953027, −6.02943072943199412125603775467, −5.02313142411893148240231572632, −3.88267212758677028526591188201, −2.69444853458753112212114778847, −2.03062707807185216085075486167, −0.71472212973628287720450526256,
0.71472212973628287720450526256, 2.03062707807185216085075486167, 2.69444853458753112212114778847, 3.88267212758677028526591188201, 5.02313142411893148240231572632, 6.02943072943199412125603775467, 6.85368690328549153891306953027, 7.74949124025404614478212183160, 8.359659388074704985756943902059, 9.188362947064529927947140817078
