| L(s) = 1 | − 2.18·2-s + 0.373·3-s + 2.76·4-s − 0.814·6-s − 2.28·7-s − 1.67·8-s − 2.86·9-s − 0.0739·11-s + 1.03·12-s − 4.15·13-s + 4.98·14-s − 1.88·16-s − 2.56·17-s + 6.24·18-s + 5.73·19-s − 0.851·21-s + 0.161·22-s − 6.89·23-s − 0.623·24-s + 9.06·26-s − 2.18·27-s − 6.31·28-s + 2.12·29-s + 1.03·31-s + 7.45·32-s − 0.0275·33-s + 5.59·34-s + ⋯ |
| L(s) = 1 | − 1.54·2-s + 0.215·3-s + 1.38·4-s − 0.332·6-s − 0.862·7-s − 0.591·8-s − 0.953·9-s − 0.0222·11-s + 0.297·12-s − 1.15·13-s + 1.33·14-s − 0.470·16-s − 0.621·17-s + 1.47·18-s + 1.31·19-s − 0.185·21-s + 0.0343·22-s − 1.43·23-s − 0.127·24-s + 1.77·26-s − 0.420·27-s − 1.19·28-s + 0.395·29-s + 0.185·31-s + 1.31·32-s − 0.00479·33-s + 0.959·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1525 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1525 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.4586359062\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.4586359062\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 \) |
| 61 | \( 1 + T \) |
| good | 2 | \( 1 + 2.18T + 2T^{2} \) |
| 3 | \( 1 - 0.373T + 3T^{2} \) |
| 7 | \( 1 + 2.28T + 7T^{2} \) |
| 11 | \( 1 + 0.0739T + 11T^{2} \) |
| 13 | \( 1 + 4.15T + 13T^{2} \) |
| 17 | \( 1 + 2.56T + 17T^{2} \) |
| 19 | \( 1 - 5.73T + 19T^{2} \) |
| 23 | \( 1 + 6.89T + 23T^{2} \) |
| 29 | \( 1 - 2.12T + 29T^{2} \) |
| 31 | \( 1 - 1.03T + 31T^{2} \) |
| 37 | \( 1 - 1.76T + 37T^{2} \) |
| 41 | \( 1 - 10.9T + 41T^{2} \) |
| 43 | \( 1 - 3.92T + 43T^{2} \) |
| 47 | \( 1 - 10.6T + 47T^{2} \) |
| 53 | \( 1 - 4.80T + 53T^{2} \) |
| 59 | \( 1 + 1.95T + 59T^{2} \) |
| 67 | \( 1 + 6.50T + 67T^{2} \) |
| 71 | \( 1 + 8.90T + 71T^{2} \) |
| 73 | \( 1 - 3.74T + 73T^{2} \) |
| 79 | \( 1 + 11.7T + 79T^{2} \) |
| 83 | \( 1 - 3.29T + 83T^{2} \) |
| 89 | \( 1 - 17.8T + 89T^{2} \) |
| 97 | \( 1 - 14.6T + 97T^{2} \) |
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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
−9.283993751166577904205399226998, −8.965676037009685441424276551754, −7.82939578864596379400126304205, −7.49087621672543384831845407530, −6.45315599577161609103086147045, −5.64560779902401647617560013544, −4.32844025452215535008861736996, −2.95274400547484548245459312333, −2.22710486939673632972177881124, −0.56570784082001056618168075524,
0.56570784082001056618168075524, 2.22710486939673632972177881124, 2.95274400547484548245459312333, 4.32844025452215535008861736996, 5.64560779902401647617560013544, 6.45315599577161609103086147045, 7.49087621672543384831845407530, 7.82939578864596379400126304205, 8.965676037009685441424276551754, 9.283993751166577904205399226998
