| L(s) = 1 | − 2-s + 0.282·3-s + 4-s + 2.81·5-s − 0.282·6-s + 7-s − 8-s − 2.92·9-s − 2.81·10-s − 3.20·11-s + 0.282·12-s + 3.10·13-s − 14-s + 0.794·15-s + 16-s − 3.75·17-s + 2.92·18-s + 2.81·20-s + 0.282·21-s + 3.20·22-s − 5.43·23-s − 0.282·24-s + 2.93·25-s − 3.10·26-s − 1.67·27-s + 28-s − 1.38·29-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 0.162·3-s + 0.5·4-s + 1.25·5-s − 0.115·6-s + 0.377·7-s − 0.353·8-s − 0.973·9-s − 0.890·10-s − 0.965·11-s + 0.0814·12-s + 0.861·13-s − 0.267·14-s + 0.205·15-s + 0.250·16-s − 0.911·17-s + 0.688·18-s + 0.629·20-s + 0.0615·21-s + 0.682·22-s − 1.13·23-s − 0.0576·24-s + 0.586·25-s − 0.609·26-s − 0.321·27-s + 0.188·28-s − 0.256·29-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5054 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5054 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.645671025\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.645671025\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 - T \) |
| 19 | \( 1 \) |
| good | 3 | \( 1 - 0.282T + 3T^{2} \) |
| 5 | \( 1 - 2.81T + 5T^{2} \) |
| 11 | \( 1 + 3.20T + 11T^{2} \) |
| 13 | \( 1 - 3.10T + 13T^{2} \) |
| 17 | \( 1 + 3.75T + 17T^{2} \) |
| 23 | \( 1 + 5.43T + 23T^{2} \) |
| 29 | \( 1 + 1.38T + 29T^{2} \) |
| 31 | \( 1 - 5.21T + 31T^{2} \) |
| 37 | \( 1 - 4.12T + 37T^{2} \) |
| 41 | \( 1 - 0.950T + 41T^{2} \) |
| 43 | \( 1 - 9.28T + 43T^{2} \) |
| 47 | \( 1 + 4.69T + 47T^{2} \) |
| 53 | \( 1 - 8.75T + 53T^{2} \) |
| 59 | \( 1 - 3.88T + 59T^{2} \) |
| 61 | \( 1 + 2.21T + 61T^{2} \) |
| 67 | \( 1 - 10.8T + 67T^{2} \) |
| 71 | \( 1 + 0.439T + 71T^{2} \) |
| 73 | \( 1 - 10.7T + 73T^{2} \) |
| 79 | \( 1 - 5.26T + 79T^{2} \) |
| 83 | \( 1 - 4.70T + 83T^{2} \) |
| 89 | \( 1 - 7.32T + 89T^{2} \) |
| 97 | \( 1 - 18.8T + 97T^{2} \) |
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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
−8.229186461877448689168057108473, −7.84825736880840116459305248830, −6.72125895717821603638801785540, −6.00113474746467460236557303677, −5.65174405129091838400866799524, −4.68099554706228583207073065836, −3.51254922178595333358504588107, −2.33926870642585444066321420424, −2.16064853242910039788305766533, −0.75061134409588181678720423345,
0.75061134409588181678720423345, 2.16064853242910039788305766533, 2.33926870642585444066321420424, 3.51254922178595333358504588107, 4.68099554706228583207073065836, 5.65174405129091838400866799524, 6.00113474746467460236557303677, 6.72125895717821603638801785540, 7.84825736880840116459305248830, 8.229186461877448689168057108473
