L(s) = 1 | + 2.73·2-s − 0.137·3-s + 5.47·4-s − 1.85·5-s − 0.375·6-s + 0.325·7-s + 9.50·8-s − 2.98·9-s − 5.06·10-s + 4.12·11-s − 0.752·12-s + 4.56·13-s + 0.888·14-s + 0.254·15-s + 15.0·16-s − 17-s − 8.15·18-s + 4.92·19-s − 10.1·20-s − 0.0446·21-s + 11.2·22-s − 5.18·23-s − 1.30·24-s − 1.56·25-s + 12.4·26-s + 0.821·27-s + 1.78·28-s + ⋯ |
L(s) = 1 | + 1.93·2-s − 0.0793·3-s + 2.73·4-s − 0.828·5-s − 0.153·6-s + 0.122·7-s + 3.35·8-s − 0.993·9-s − 1.60·10-s + 1.24·11-s − 0.217·12-s + 1.26·13-s + 0.237·14-s + 0.0657·15-s + 3.75·16-s − 0.242·17-s − 1.92·18-s + 1.13·19-s − 2.26·20-s − 0.00974·21-s + 2.40·22-s − 1.08·23-s − 0.266·24-s − 0.313·25-s + 2.44·26-s + 0.158·27-s + 0.336·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1003 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1003 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(4.803160479\) |
\(L(\frac12)\) |
\(\approx\) |
\(4.803160479\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 17 | \( 1 + T \) |
| 59 | \( 1 - T \) |
good | 2 | \( 1 - 2.73T + 2T^{2} \) |
| 3 | \( 1 + 0.137T + 3T^{2} \) |
| 5 | \( 1 + 1.85T + 5T^{2} \) |
| 7 | \( 1 - 0.325T + 7T^{2} \) |
| 11 | \( 1 - 4.12T + 11T^{2} \) |
| 13 | \( 1 - 4.56T + 13T^{2} \) |
| 19 | \( 1 - 4.92T + 19T^{2} \) |
| 23 | \( 1 + 5.18T + 23T^{2} \) |
| 29 | \( 1 + 6.24T + 29T^{2} \) |
| 31 | \( 1 - 2.75T + 31T^{2} \) |
| 37 | \( 1 + 4.40T + 37T^{2} \) |
| 41 | \( 1 - 3.43T + 41T^{2} \) |
| 43 | \( 1 + 8.65T + 43T^{2} \) |
| 47 | \( 1 - 12.5T + 47T^{2} \) |
| 53 | \( 1 - 0.346T + 53T^{2} \) |
| 61 | \( 1 + 15.1T + 61T^{2} \) |
| 67 | \( 1 + 13.1T + 67T^{2} \) |
| 71 | \( 1 - 2.72T + 71T^{2} \) |
| 73 | \( 1 + 11.0T + 73T^{2} \) |
| 79 | \( 1 + 5.84T + 79T^{2} \) |
| 83 | \( 1 + 5.47T + 83T^{2} \) |
| 89 | \( 1 - 0.986T + 89T^{2} \) |
| 97 | \( 1 - 0.328T + 97T^{2} \) |
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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
−10.46527194659563753508319182785, −9.020645753469281550918458695523, −7.980425921781630760533317538825, −7.17183158861522234928471315581, −6.12706737142108097784746687214, −5.74249302661817703229764405104, −4.51214634664209440729686094739, −3.75528004823199542073060463620, −3.16512440474011427721789398826, −1.64248954246910494096196689379,
1.64248954246910494096196689379, 3.16512440474011427721789398826, 3.75528004823199542073060463620, 4.51214634664209440729686094739, 5.74249302661817703229764405104, 6.12706737142108097784746687214, 7.17183158861522234928471315581, 7.980425921781630760533317538825, 9.020645753469281550918458695523, 10.46527194659563753508319182785