| L(s) = 1 | + 0.517·2-s − 1.93·3-s − 1.73·4-s − 0.999·6-s − 3.34·7-s − 1.93·8-s + 0.732·9-s + 3.34·12-s + 4.24·13-s − 1.73·14-s + 2.46·16-s + 3.86·17-s + 0.378·18-s − 4.19·19-s + 6.46·21-s + 3.48·23-s + 3.73·24-s + 2.19·26-s + 4.38·27-s + 5.79·28-s + 6.92·29-s − 8.73·31-s + 5.13·32-s + 1.99·34-s − 1.26·36-s + 1.79·37-s − 2.17·38-s + ⋯ |
| L(s) = 1 | + 0.366·2-s − 1.11·3-s − 0.866·4-s − 0.408·6-s − 1.26·7-s − 0.683·8-s + 0.244·9-s + 0.965·12-s + 1.17·13-s − 0.462·14-s + 0.616·16-s + 0.937·17-s + 0.0893·18-s − 0.962·19-s + 1.41·21-s + 0.726·23-s + 0.761·24-s + 0.430·26-s + 0.843·27-s + 1.09·28-s + 1.28·29-s − 1.56·31-s + 0.908·32-s + 0.342·34-s − 0.211·36-s + 0.294·37-s − 0.352·38-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3025 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3025 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 \) |
| 11 | \( 1 \) |
| good | 2 | \( 1 - 0.517T + 2T^{2} \) |
| 3 | \( 1 + 1.93T + 3T^{2} \) |
| 7 | \( 1 + 3.34T + 7T^{2} \) |
| 13 | \( 1 - 4.24T + 13T^{2} \) |
| 17 | \( 1 - 3.86T + 17T^{2} \) |
| 19 | \( 1 + 4.19T + 19T^{2} \) |
| 23 | \( 1 - 3.48T + 23T^{2} \) |
| 29 | \( 1 - 6.92T + 29T^{2} \) |
| 31 | \( 1 + 8.73T + 31T^{2} \) |
| 37 | \( 1 - 1.79T + 37T^{2} \) |
| 41 | \( 1 - 1.73T + 41T^{2} \) |
| 43 | \( 1 + 6.45T + 43T^{2} \) |
| 47 | \( 1 - 11.4T + 47T^{2} \) |
| 53 | \( 1 - 2.17T + 53T^{2} \) |
| 59 | \( 1 + 1.26T + 59T^{2} \) |
| 61 | \( 1 + 11.7T + 61T^{2} \) |
| 67 | \( 1 - 2.20T + 67T^{2} \) |
| 71 | \( 1 + 8.19T + 71T^{2} \) |
| 73 | \( 1 + 4.89T + 73T^{2} \) |
| 79 | \( 1 - 1.46T + 79T^{2} \) |
| 83 | \( 1 + 9.89T + 83T^{2} \) |
| 89 | \( 1 - 0.464T + 89T^{2} \) |
| 97 | \( 1 + 9.14T + 97T^{2} \) |
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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
−8.654204565543805455541890313231, −7.43803142534412516689645547477, −6.41783765621571101022801522029, −6.03441570535166690865014845829, −5.40280877030007592551140254513, −4.51088373721475867260674830648, −3.64835544944971181545803879043, −2.95012040401598536026634142322, −1.05756869788692709244837772831, 0,
1.05756869788692709244837772831, 2.95012040401598536026634142322, 3.64835544944971181545803879043, 4.51088373721475867260674830648, 5.40280877030007592551140254513, 6.03441570535166690865014845829, 6.41783765621571101022801522029, 7.43803142534412516689645547477, 8.654204565543805455541890313231
