| L(s) = 1 | − 2.19·2-s + 1.59·3-s + 2.81·4-s − 3.50·6-s + 3.21·7-s − 1.79·8-s − 0.444·9-s + 1.61·11-s + 4.50·12-s − 6.08·13-s − 7.05·14-s − 1.69·16-s + 0.975·18-s + 3.45·19-s + 5.13·21-s − 3.54·22-s + 6.00·23-s − 2.86·24-s + 13.3·26-s − 5.50·27-s + 9.05·28-s + 3.36·29-s + 4.00·31-s + 7.31·32-s + 2.58·33-s − 1.25·36-s + 8.05·37-s + ⋯ |
| L(s) = 1 | − 1.55·2-s + 0.922·3-s + 1.40·4-s − 1.43·6-s + 1.21·7-s − 0.634·8-s − 0.148·9-s + 0.487·11-s + 1.30·12-s − 1.68·13-s − 1.88·14-s − 0.424·16-s + 0.229·18-s + 0.792·19-s + 1.12·21-s − 0.755·22-s + 1.25·23-s − 0.585·24-s + 2.61·26-s − 1.05·27-s + 1.71·28-s + 0.624·29-s + 0.719·31-s + 1.29·32-s + 0.449·33-s − 0.208·36-s + 1.32·37-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 7225 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7225 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.440852304\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.440852304\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 \) |
| 17 | \( 1 \) |
| good | 2 | \( 1 + 2.19T + 2T^{2} \) |
| 3 | \( 1 - 1.59T + 3T^{2} \) |
| 7 | \( 1 - 3.21T + 7T^{2} \) |
| 11 | \( 1 - 1.61T + 11T^{2} \) |
| 13 | \( 1 + 6.08T + 13T^{2} \) |
| 19 | \( 1 - 3.45T + 19T^{2} \) |
| 23 | \( 1 - 6.00T + 23T^{2} \) |
| 29 | \( 1 - 3.36T + 29T^{2} \) |
| 31 | \( 1 - 4.00T + 31T^{2} \) |
| 37 | \( 1 - 8.05T + 37T^{2} \) |
| 41 | \( 1 + 8.41T + 41T^{2} \) |
| 43 | \( 1 - 12.9T + 43T^{2} \) |
| 47 | \( 1 + 2.53T + 47T^{2} \) |
| 53 | \( 1 + 4.83T + 53T^{2} \) |
| 59 | \( 1 - 11.3T + 59T^{2} \) |
| 61 | \( 1 + 5.48T + 61T^{2} \) |
| 67 | \( 1 + 8.63T + 67T^{2} \) |
| 71 | \( 1 + 10.9T + 71T^{2} \) |
| 73 | \( 1 + 15.3T + 73T^{2} \) |
| 79 | \( 1 - 2.08T + 79T^{2} \) |
| 83 | \( 1 - 15.0T + 83T^{2} \) |
| 89 | \( 1 + 4.63T + 89T^{2} \) |
| 97 | \( 1 - 8.44T + 97T^{2} \) |
| show more | |
| show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−7.961601892748421928631000517784, −7.52969255689496208425472585255, −7.11120811189929948520558196577, −6.03190753099388471072938683118, −4.92254575858430339739552574911, −4.48079695980548683572162671435, −3.07943142920212847478775611152, −2.48464900444980823199480819317, −1.67727477703597286778176146151, −0.75678121820611436084272475191,
0.75678121820611436084272475191, 1.67727477703597286778176146151, 2.48464900444980823199480819317, 3.07943142920212847478775611152, 4.48079695980548683572162671435, 4.92254575858430339739552574911, 6.03190753099388471072938683118, 7.11120811189929948520558196577, 7.52969255689496208425472585255, 7.961601892748421928631000517784
