L(s) = 1 | − 0.894·2-s + 2.74·3-s − 1.19·4-s − 2.45·6-s + 7-s + 2.86·8-s + 4.55·9-s + 0.105·11-s − 3.29·12-s + 5.45·13-s − 0.894·14-s − 0.160·16-s + 0.992·17-s − 4.07·18-s + 3.34·19-s + 2.74·21-s − 0.0944·22-s − 23-s + 7.86·24-s − 4.87·26-s + 4.27·27-s − 1.19·28-s + 4.90·29-s + 4.24·31-s − 5.58·32-s + 0.290·33-s − 0.887·34-s + ⋯ |
L(s) = 1 | − 0.632·2-s + 1.58·3-s − 0.599·4-s − 1.00·6-s + 0.377·7-s + 1.01·8-s + 1.51·9-s + 0.0318·11-s − 0.952·12-s + 1.51·13-s − 0.239·14-s − 0.0400·16-s + 0.240·17-s − 0.960·18-s + 0.767·19-s + 0.599·21-s − 0.0201·22-s − 0.208·23-s + 1.60·24-s − 0.956·26-s + 0.823·27-s − 0.226·28-s + 0.910·29-s + 0.762·31-s − 0.986·32-s + 0.0505·33-s − 0.152·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4025 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4025 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.578513066\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.578513066\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 5 | \( 1 \) |
| 7 | \( 1 - T \) |
| 23 | \( 1 + T \) |
good | 2 | \( 1 + 0.894T + 2T^{2} \) |
| 3 | \( 1 - 2.74T + 3T^{2} \) |
| 11 | \( 1 - 0.105T + 11T^{2} \) |
| 13 | \( 1 - 5.45T + 13T^{2} \) |
| 17 | \( 1 - 0.992T + 17T^{2} \) |
| 19 | \( 1 - 3.34T + 19T^{2} \) |
| 29 | \( 1 - 4.90T + 29T^{2} \) |
| 31 | \( 1 - 4.24T + 31T^{2} \) |
| 37 | \( 1 + 4.79T + 37T^{2} \) |
| 41 | \( 1 + 4.65T + 41T^{2} \) |
| 43 | \( 1 - 1.03T + 43T^{2} \) |
| 47 | \( 1 - 0.654T + 47T^{2} \) |
| 53 | \( 1 + 6.49T + 53T^{2} \) |
| 59 | \( 1 - 4.54T + 59T^{2} \) |
| 61 | \( 1 - 1.98T + 61T^{2} \) |
| 67 | \( 1 + 7.83T + 67T^{2} \) |
| 71 | \( 1 + 7.53T + 71T^{2} \) |
| 73 | \( 1 - 14.6T + 73T^{2} \) |
| 79 | \( 1 - 2.59T + 79T^{2} \) |
| 83 | \( 1 + 1.14T + 83T^{2} \) |
| 89 | \( 1 + 14.3T + 89T^{2} \) |
| 97 | \( 1 - 13.9T + 97T^{2} \) |
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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
−8.540603537045510415577394386522, −8.026177014470209206673372556359, −7.42022482775475654181922719695, −6.45808323156312496164265978916, −5.34320124305773385023476540385, −4.43008021996689258949942614971, −3.69272102435647643107856503564, −3.03055967876254221737300061797, −1.81187115503270364361181325845, −1.04223558784363552025367614830,
1.04223558784363552025367614830, 1.81187115503270364361181325845, 3.03055967876254221737300061797, 3.69272102435647643107856503564, 4.43008021996689258949942614971, 5.34320124305773385023476540385, 6.45808323156312496164265978916, 7.42022482775475654181922719695, 8.026177014470209206673372556359, 8.540603537045510415577394386522