| L(s) = 1 | + 1.69·2-s − 3-s + 0.882·4-s + 1.54·5-s − 1.69·6-s − 7-s − 1.89·8-s + 9-s + 2.61·10-s + 6.17·11-s − 0.882·12-s + 0.0802·13-s − 1.69·14-s − 1.54·15-s − 4.98·16-s − 2.27·17-s + 1.69·18-s + 1.09·19-s + 1.36·20-s + 21-s + 10.4·22-s + 8.34·23-s + 1.89·24-s − 2.62·25-s + 0.136·26-s − 27-s − 0.882·28-s + ⋯ |
| L(s) = 1 | + 1.20·2-s − 0.577·3-s + 0.441·4-s + 0.689·5-s − 0.693·6-s − 0.377·7-s − 0.670·8-s + 0.333·9-s + 0.828·10-s + 1.86·11-s − 0.254·12-s + 0.0222·13-s − 0.453·14-s − 0.398·15-s − 1.24·16-s − 0.550·17-s + 0.400·18-s + 0.251·19-s + 0.304·20-s + 0.218·21-s + 2.23·22-s + 1.73·23-s + 0.387·24-s − 0.524·25-s + 0.0267·26-s − 0.192·27-s − 0.166·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8043 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8043 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.527085496\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.527085496\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + T \) |
| 7 | \( 1 + T \) |
| 383 | \( 1 - T \) |
| good | 2 | \( 1 - 1.69T + 2T^{2} \) |
| 5 | \( 1 - 1.54T + 5T^{2} \) |
| 11 | \( 1 - 6.17T + 11T^{2} \) |
| 13 | \( 1 - 0.0802T + 13T^{2} \) |
| 17 | \( 1 + 2.27T + 17T^{2} \) |
| 19 | \( 1 - 1.09T + 19T^{2} \) |
| 23 | \( 1 - 8.34T + 23T^{2} \) |
| 29 | \( 1 + 6.73T + 29T^{2} \) |
| 31 | \( 1 - 1.84T + 31T^{2} \) |
| 37 | \( 1 - 1.31T + 37T^{2} \) |
| 41 | \( 1 - 7.26T + 41T^{2} \) |
| 43 | \( 1 - 3.57T + 43T^{2} \) |
| 47 | \( 1 - 7.08T + 47T^{2} \) |
| 53 | \( 1 + 4.53T + 53T^{2} \) |
| 59 | \( 1 - 10.4T + 59T^{2} \) |
| 61 | \( 1 + 9.05T + 61T^{2} \) |
| 67 | \( 1 + 0.858T + 67T^{2} \) |
| 71 | \( 1 + 10.5T + 71T^{2} \) |
| 73 | \( 1 + 8.65T + 73T^{2} \) |
| 79 | \( 1 - 7.62T + 79T^{2} \) |
| 83 | \( 1 - 3.36T + 83T^{2} \) |
| 89 | \( 1 - 6.35T + 89T^{2} \) |
| 97 | \( 1 - 1.68T + 97T^{2} \) |
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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
−7.39650534751071520360200184193, −6.85651186217150463712298971509, −6.06645246849334530332279898397, −5.92629561571310381946511119618, −5.00770998783882282353390491409, −4.30844537240203596420202971604, −3.75113775054073105620558703732, −2.90980551963482906167725917486, −1.88547649841641254154858555623, −0.816853775125628154267320949143,
0.816853775125628154267320949143, 1.88547649841641254154858555623, 2.90980551963482906167725917486, 3.75113775054073105620558703732, 4.30844537240203596420202971604, 5.00770998783882282353390491409, 5.92629561571310381946511119618, 6.06645246849334530332279898397, 6.85651186217150463712298971509, 7.39650534751071520360200184193
