| L(s) = 1 | + 2-s + 0.395·3-s + 4-s + 0.151·5-s + 0.395·6-s − 7-s + 8-s − 2.84·9-s + 0.151·10-s + 2.85·11-s + 0.395·12-s − 2.70·13-s − 14-s + 0.0600·15-s + 16-s − 1.00·17-s − 2.84·18-s + 2.34·19-s + 0.151·20-s − 0.395·21-s + 2.85·22-s − 0.830·23-s + 0.395·24-s − 4.97·25-s − 2.70·26-s − 2.31·27-s − 28-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 0.228·3-s + 0.5·4-s + 0.0678·5-s + 0.161·6-s − 0.377·7-s + 0.353·8-s − 0.947·9-s + 0.0479·10-s + 0.859·11-s + 0.114·12-s − 0.751·13-s − 0.267·14-s + 0.0155·15-s + 0.250·16-s − 0.243·17-s − 0.670·18-s + 0.538·19-s + 0.0339·20-s − 0.0863·21-s + 0.607·22-s − 0.173·23-s + 0.0808·24-s − 0.995·25-s − 0.531·26-s − 0.445·27-s − 0.188·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6034 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6034 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 2 | \( 1 - T \) |
| 7 | \( 1 + T \) |
| 431 | \( 1 - T \) |
| good | 3 | \( 1 - 0.395T + 3T^{2} \) |
| 5 | \( 1 - 0.151T + 5T^{2} \) |
| 11 | \( 1 - 2.85T + 11T^{2} \) |
| 13 | \( 1 + 2.70T + 13T^{2} \) |
| 17 | \( 1 + 1.00T + 17T^{2} \) |
| 19 | \( 1 - 2.34T + 19T^{2} \) |
| 23 | \( 1 + 0.830T + 23T^{2} \) |
| 29 | \( 1 - 3.14T + 29T^{2} \) |
| 31 | \( 1 + 4.76T + 31T^{2} \) |
| 37 | \( 1 - 2.32T + 37T^{2} \) |
| 41 | \( 1 + 3.35T + 41T^{2} \) |
| 43 | \( 1 + 11.1T + 43T^{2} \) |
| 47 | \( 1 + 9.19T + 47T^{2} \) |
| 53 | \( 1 - 0.902T + 53T^{2} \) |
| 59 | \( 1 + 0.539T + 59T^{2} \) |
| 61 | \( 1 + 3.28T + 61T^{2} \) |
| 67 | \( 1 + 8.88T + 67T^{2} \) |
| 71 | \( 1 - 9.68T + 71T^{2} \) |
| 73 | \( 1 - 1.66T + 73T^{2} \) |
| 79 | \( 1 - 4.67T + 79T^{2} \) |
| 83 | \( 1 - 13.9T + 83T^{2} \) |
| 89 | \( 1 + 1.06T + 89T^{2} \) |
| 97 | \( 1 + 6.33T + 97T^{2} \) |
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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
−7.73132547177836330592270320872, −6.76233694990038889352468436575, −6.36296139973660830877738842240, −5.48466356314661898011966006153, −4.91620876500919745965232446533, −3.90860292364024532773493676157, −3.31304534987949798704072943927, −2.51276360411671158116406788945, −1.59131687563656190987979085415, 0,
1.59131687563656190987979085415, 2.51276360411671158116406788945, 3.31304534987949798704072943927, 3.90860292364024532773493676157, 4.91620876500919745965232446533, 5.48466356314661898011966006153, 6.36296139973660830877738842240, 6.76233694990038889352468436575, 7.73132547177836330592270320872
