| L(s) = 1 | + 2.41·2-s + 3.82·4-s + 7-s + 4.41·8-s + 0.828·11-s + 4.82·13-s + 2.41·14-s + 2.99·16-s − 3.65·17-s + 2.82·19-s + 1.99·22-s + 7.65·23-s + 11.6·26-s + 3.82·28-s − 8·29-s − 8.48·31-s − 1.58·32-s − 8.82·34-s + 6·37-s + 6.82·38-s − 3.65·41-s + 9.65·43-s + 3.17·44-s + 18.4·46-s − 4·47-s + 49-s + 18.4·52-s + ⋯ |
| L(s) = 1 | + 1.70·2-s + 1.91·4-s + 0.377·7-s + 1.56·8-s + 0.249·11-s + 1.33·13-s + 0.645·14-s + 0.749·16-s − 0.886·17-s + 0.648·19-s + 0.426·22-s + 1.59·23-s + 2.28·26-s + 0.723·28-s − 1.48·29-s − 1.52·31-s − 0.280·32-s − 1.51·34-s + 0.986·37-s + 1.10·38-s − 0.571·41-s + 1.47·43-s + 0.478·44-s + 2.72·46-s − 0.583·47-s + 0.142·49-s + 2.56·52-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1575 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1575 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(5.092826665\) |
| \(L(\frac12)\) |
\(\approx\) |
\(5.092826665\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 5 | \( 1 \) |
| 7 | \( 1 - T \) |
| good | 2 | \( 1 - 2.41T + 2T^{2} \) |
| 11 | \( 1 - 0.828T + 11T^{2} \) |
| 13 | \( 1 - 4.82T + 13T^{2} \) |
| 17 | \( 1 + 3.65T + 17T^{2} \) |
| 19 | \( 1 - 2.82T + 19T^{2} \) |
| 23 | \( 1 - 7.65T + 23T^{2} \) |
| 29 | \( 1 + 8T + 29T^{2} \) |
| 31 | \( 1 + 8.48T + 31T^{2} \) |
| 37 | \( 1 - 6T + 37T^{2} \) |
| 41 | \( 1 + 3.65T + 41T^{2} \) |
| 43 | \( 1 - 9.65T + 43T^{2} \) |
| 47 | \( 1 + 4T + 47T^{2} \) |
| 53 | \( 1 - 10.8T + 53T^{2} \) |
| 59 | \( 1 - 4T + 59T^{2} \) |
| 61 | \( 1 - 6T + 61T^{2} \) |
| 67 | \( 1 + 7.31T + 67T^{2} \) |
| 71 | \( 1 - 6.48T + 71T^{2} \) |
| 73 | \( 1 + 16.1T + 73T^{2} \) |
| 79 | \( 1 + 5.65T + 79T^{2} \) |
| 83 | \( 1 + 8T + 83T^{2} \) |
| 89 | \( 1 + 17.3T + 89T^{2} \) |
| 97 | \( 1 - 8.82T + 97T^{2} \) |
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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
−9.300043037037285924493162832455, −8.672139348674568336472164397788, −7.39917906707611315601409035028, −6.84874101280853142839110277633, −5.81909766951423116864094607071, −5.36203074373124717314864120749, −4.28388104909836250279335960684, −3.69680734059705540929043331236, −2.69623861641071732260051546833, −1.48755067666869266520752626945,
1.48755067666869266520752626945, 2.69623861641071732260051546833, 3.69680734059705540929043331236, 4.28388104909836250279335960684, 5.36203074373124717314864120749, 5.81909766951423116864094607071, 6.84874101280853142839110277633, 7.39917906707611315601409035028, 8.672139348674568336472164397788, 9.300043037037285924493162832455
