L(s) = 1 | − 2-s − 1.61·3-s + 4-s + 5-s + 1.61·6-s − 7-s − 8-s − 0.381·9-s − 10-s − 5·11-s − 1.61·12-s + 1.85·13-s + 14-s − 1.61·15-s + 16-s + 0.145·17-s + 0.381·18-s + 20-s + 1.61·21-s + 5·22-s − 8.23·23-s + 1.61·24-s − 4·25-s − 1.85·26-s + 5.47·27-s − 28-s + 3.76·29-s + ⋯ |
L(s) = 1 | − 0.707·2-s − 0.934·3-s + 0.5·4-s + 0.447·5-s + 0.660·6-s − 0.377·7-s − 0.353·8-s − 0.127·9-s − 0.316·10-s − 1.50·11-s − 0.467·12-s + 0.514·13-s + 0.267·14-s − 0.417·15-s + 0.250·16-s + 0.0353·17-s + 0.0900·18-s + 0.223·20-s + 0.353·21-s + 1.06·22-s − 1.71·23-s + 0.330·24-s − 0.800·25-s − 0.363·26-s + 1.05·27-s − 0.188·28-s + 0.698·29-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
\(L(1)\) |
\(\approx\) |
\(0.3977880872\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.3977880872\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 + 1.61T + 3T^{2} \) |
| 5 | \( 1 - T + 5T^{2} \) |
| 11 | \( 1 + 5T + 11T^{2} \) |
| 13 | \( 1 - 1.85T + 13T^{2} \) |
| 17 | \( 1 - 0.145T + 17T^{2} \) |
| 23 | \( 1 + 8.23T + 23T^{2} \) |
| 29 | \( 1 - 3.76T + 29T^{2} \) |
| 31 | \( 1 + 6.70T + 31T^{2} \) |
| 37 | \( 1 - 11.9T + 37T^{2} \) |
| 41 | \( 1 + 10.0T + 41T^{2} \) |
| 43 | \( 1 + 0.145T + 43T^{2} \) |
| 47 | \( 1 + 9.94T + 47T^{2} \) |
| 53 | \( 1 + 0.708T + 53T^{2} \) |
| 59 | \( 1 + 11T + 59T^{2} \) |
| 61 | \( 1 + 1.47T + 61T^{2} \) |
| 67 | \( 1 + 1.14T + 67T^{2} \) |
| 71 | \( 1 - 10T + 71T^{2} \) |
| 73 | \( 1 - 5.56T + 73T^{2} \) |
| 79 | \( 1 - 2.14T + 79T^{2} \) |
| 83 | \( 1 - 11.4T + 83T^{2} \) |
| 89 | \( 1 - 1.85T + 89T^{2} \) |
| 97 | \( 1 - 0.527T + 97T^{2} \) |
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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
−7.986933398573022954631148065943, −7.83625840071520840875154358713, −6.55979707674664762808777270128, −6.15502733992582116059448380318, −5.53132318283065462015926133224, −4.82763886123802459223707269336, −3.60541755521651969457940174372, −2.64724123849822413007026068058, −1.77703535983701164350989864319, −0.38431153708137697489442842271,
0.38431153708137697489442842271, 1.77703535983701164350989864319, 2.64724123849822413007026068058, 3.60541755521651969457940174372, 4.82763886123802459223707269336, 5.53132318283065462015926133224, 6.15502733992582116059448380318, 6.55979707674664762808777270128, 7.83625840071520840875154358713, 7.986933398573022954631148065943