| L(s) = 1 | − 0.752·2-s − 2.90·3-s − 1.43·4-s + 2.18·6-s + 2.58·8-s + 5.43·9-s − 11-s + 4.16·12-s + 0.470·13-s + 0.921·16-s − 17-s − 4.09·18-s − 6.84·19-s + 0.752·22-s + 2.40·23-s − 7.50·24-s − 0.354·26-s − 7.06·27-s − 0.186·29-s + 7.93·31-s − 5.86·32-s + 2.90·33-s + 0.752·34-s − 7.78·36-s − 8.40·37-s + 5.15·38-s − 1.36·39-s + ⋯ |
| L(s) = 1 | − 0.532·2-s − 1.67·3-s − 0.716·4-s + 0.892·6-s + 0.913·8-s + 1.81·9-s − 0.301·11-s + 1.20·12-s + 0.130·13-s + 0.230·16-s − 0.242·17-s − 0.964·18-s − 1.56·19-s + 0.160·22-s + 0.502·23-s − 1.53·24-s − 0.0694·26-s − 1.35·27-s − 0.0345·29-s + 1.42·31-s − 1.03·32-s + 0.505·33-s + 0.129·34-s − 1.29·36-s − 1.38·37-s + 0.835·38-s − 0.218·39-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4675 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4675 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 17 | \( 1 + T \) |
| good | 2 | \( 1 + 0.752T + 2T^{2} \) |
| 3 | \( 1 + 2.90T + 3T^{2} \) |
| 7 | \( 1 + 7T^{2} \) |
| 13 | \( 1 - 0.470T + 13T^{2} \) |
| 19 | \( 1 + 6.84T + 19T^{2} \) |
| 23 | \( 1 - 2.40T + 23T^{2} \) |
| 29 | \( 1 + 0.186T + 29T^{2} \) |
| 31 | \( 1 - 7.93T + 31T^{2} \) |
| 37 | \( 1 + 8.40T + 37T^{2} \) |
| 41 | \( 1 + 3.80T + 41T^{2} \) |
| 43 | \( 1 - 0.941T + 43T^{2} \) |
| 47 | \( 1 - 1.15T + 47T^{2} \) |
| 53 | \( 1 + 4.84T + 53T^{2} \) |
| 59 | \( 1 - 2.13T + 59T^{2} \) |
| 61 | \( 1 - 5.01T + 61T^{2} \) |
| 67 | \( 1 - 8.10T + 67T^{2} \) |
| 71 | \( 1 - 13.4T + 71T^{2} \) |
| 73 | \( 1 - 12.0T + 73T^{2} \) |
| 79 | \( 1 + 1.36T + 79T^{2} \) |
| 83 | \( 1 + 16.6T + 83T^{2} \) |
| 89 | \( 1 - 16.2T + 89T^{2} \) |
| 97 | \( 1 + 11.2T + 97T^{2} \) |
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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
−8.123389329924086501532429865794, −6.99490332762327810247795461573, −6.54837308883844940385835245208, −5.72015239671027555017092251789, −4.92407137517981609352075748643, −4.56570385654957578655742580379, −3.59562924370034256149020679513, −2.02085434113379016765662616128, −0.893574738788927612819947501043, 0,
0.893574738788927612819947501043, 2.02085434113379016765662616128, 3.59562924370034256149020679513, 4.56570385654957578655742580379, 4.92407137517981609352075748643, 5.72015239671027555017092251789, 6.54837308883844940385835245208, 6.99490332762327810247795461573, 8.123389329924086501532429865794
