| L(s) = 1 | + 0.0377·2-s − 2.71·3-s − 1.99·4-s − 0.102·6-s − 0.366·7-s − 0.150·8-s + 4.38·9-s + 1.43·11-s + 5.42·12-s + 4.61·13-s − 0.0138·14-s + 3.99·16-s + 0.165·18-s − 3.66·19-s + 0.994·21-s + 0.0543·22-s + 8.56·23-s + 0.409·24-s + 0.174·26-s − 3.75·27-s + 0.731·28-s + 8.61·29-s + 9.41·31-s + 0.452·32-s − 3.91·33-s − 8.75·36-s + 7.58·37-s + ⋯ |
| L(s) = 1 | + 0.0266·2-s − 1.56·3-s − 0.999·4-s − 0.0418·6-s − 0.138·7-s − 0.0533·8-s + 1.46·9-s + 0.434·11-s + 1.56·12-s + 1.27·13-s − 0.00368·14-s + 0.997·16-s + 0.0389·18-s − 0.841·19-s + 0.217·21-s + 0.0115·22-s + 1.78·23-s + 0.0836·24-s + 0.0341·26-s − 0.722·27-s + 0.138·28-s + 1.60·29-s + 1.69·31-s + 0.0799·32-s − 0.681·33-s − 1.45·36-s + 1.24·37-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
| \(L(1)\) |
\(\approx\) |
\(1.087407323\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.087407323\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 - 0.0377T + 2T^{2} \) |
| 3 | \( 1 + 2.71T + 3T^{2} \) |
| 7 | \( 1 + 0.366T + 7T^{2} \) |
| 11 | \( 1 - 1.43T + 11T^{2} \) |
| 13 | \( 1 - 4.61T + 13T^{2} \) |
| 19 | \( 1 + 3.66T + 19T^{2} \) |
| 23 | \( 1 - 8.56T + 23T^{2} \) |
| 29 | \( 1 - 8.61T + 29T^{2} \) |
| 31 | \( 1 - 9.41T + 31T^{2} \) |
| 37 | \( 1 - 7.58T + 37T^{2} \) |
| 41 | \( 1 + 6.18T + 41T^{2} \) |
| 43 | \( 1 - 7.04T + 43T^{2} \) |
| 47 | \( 1 + 3.75T + 47T^{2} \) |
| 53 | \( 1 - 10.3T + 53T^{2} \) |
| 59 | \( 1 - 6.93T + 59T^{2} \) |
| 61 | \( 1 - 2.29T + 61T^{2} \) |
| 67 | \( 1 + 5.77T + 67T^{2} \) |
| 71 | \( 1 - 6.76T + 71T^{2} \) |
| 73 | \( 1 - 4.33T + 73T^{2} \) |
| 79 | \( 1 - 4.06T + 79T^{2} \) |
| 83 | \( 1 - 11.6T + 83T^{2} \) |
| 89 | \( 1 + 10.6T + 89T^{2} \) |
| 97 | \( 1 - 8.52T + 97T^{2} \) |
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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
−8.100035470018131878937588460490, −6.78669914045574447021627532301, −6.49798380518576908138647843722, −5.82227610327888668626337695814, −5.05179561975298486598099819648, −4.53809728863019964140760652922, −3.88076114589475993880832346531, −2.84893419611356005748497592989, −1.15312957064579406197223828476, −0.73413163314516577885381582160,
0.73413163314516577885381582160, 1.15312957064579406197223828476, 2.84893419611356005748497592989, 3.88076114589475993880832346531, 4.53809728863019964140760652922, 5.05179561975298486598099819648, 5.82227610327888668626337695814, 6.49798380518576908138647843722, 6.78669914045574447021627532301, 8.100035470018131878937588460490
