| L(s) = 1 | − 2-s − 1.41·3-s + 4-s + 1.41·6-s + 7-s − 8-s − 0.999·9-s − 11-s − 1.41·12-s + 5.24·13-s − 14-s + 16-s + 5.41·17-s + 0.999·18-s + 5.82·19-s − 1.41·21-s + 22-s + 4.41·23-s + 1.41·24-s − 5.24·26-s + 5.65·27-s + 28-s − 2.65·29-s − 5.82·31-s − 32-s + 1.41·33-s − 5.41·34-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 0.816·3-s + 0.5·4-s + 0.577·6-s + 0.377·7-s − 0.353·8-s − 0.333·9-s − 0.301·11-s − 0.408·12-s + 1.45·13-s − 0.267·14-s + 0.250·16-s + 1.31·17-s + 0.235·18-s + 1.33·19-s − 0.308·21-s + 0.213·22-s + 0.920·23-s + 0.288·24-s − 1.02·26-s + 1.08·27-s + 0.188·28-s − 0.493·29-s − 1.04·31-s − 0.176·32-s + 0.246·33-s − 0.928·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3850 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3850 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.107429155\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.107429155\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + T \) |
| 5 | \( 1 \) |
| 7 | \( 1 - T \) |
| 11 | \( 1 + T \) |
| good | 3 | \( 1 + 1.41T + 3T^{2} \) |
| 13 | \( 1 - 5.24T + 13T^{2} \) |
| 17 | \( 1 - 5.41T + 17T^{2} \) |
| 19 | \( 1 - 5.82T + 19T^{2} \) |
| 23 | \( 1 - 4.41T + 23T^{2} \) |
| 29 | \( 1 + 2.65T + 29T^{2} \) |
| 31 | \( 1 + 5.82T + 31T^{2} \) |
| 37 | \( 1 + 2.82T + 37T^{2} \) |
| 41 | \( 1 + 1.17T + 41T^{2} \) |
| 43 | \( 1 + 8.89T + 43T^{2} \) |
| 47 | \( 1 - 10.4T + 47T^{2} \) |
| 53 | \( 1 - 3.75T + 53T^{2} \) |
| 59 | \( 1 - 4T + 59T^{2} \) |
| 61 | \( 1 + 7.17T + 61T^{2} \) |
| 67 | \( 1 - 0.585T + 67T^{2} \) |
| 71 | \( 1 + 4.07T + 71T^{2} \) |
| 73 | \( 1 + 7.31T + 73T^{2} \) |
| 79 | \( 1 + 3.07T + 79T^{2} \) |
| 83 | \( 1 - 8.17T + 83T^{2} \) |
| 89 | \( 1 - 12.8T + 89T^{2} \) |
| 97 | \( 1 + 12.0T + 97T^{2} \) |
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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
−8.599345476746735520306071270514, −7.70580534379041654643466395251, −7.17402010144110363659786148482, −6.18440349075900501123487955172, −5.55413364143921946181724119381, −5.09470622892454678601563370249, −3.67103822459848905940435124397, −3.00339747590084227626455225710, −1.56534718280333552645354523785, −0.76043492126337774968401401519,
0.76043492126337774968401401519, 1.56534718280333552645354523785, 3.00339747590084227626455225710, 3.67103822459848905940435124397, 5.09470622892454678601563370249, 5.55413364143921946181724119381, 6.18440349075900501123487955172, 7.17402010144110363659786148482, 7.70580534379041654643466395251, 8.599345476746735520306071270514
