L(s) = 1 | + 1.94·2-s − 1.49·3-s + 2.77·4-s − 5-s − 2.90·6-s − 1.13·7-s + 3.43·8-s + 1.24·9-s − 1.94·10-s + 0.709·11-s − 4.14·12-s + 13-s − 2.20·14-s + 1.49·15-s + 3.90·16-s + 1.77·17-s + 2.41·18-s − 2.77·20-s + 1.70·21-s + 1.37·22-s + 0.241·23-s − 5.14·24-s + 25-s + 1.94·26-s − 0.360·27-s − 3.14·28-s + 2.90·30-s + ⋯ |
L(s) = 1 | + 1.94·2-s − 1.49·3-s + 2.77·4-s − 5-s − 2.90·6-s − 1.13·7-s + 3.43·8-s + 1.24·9-s − 1.94·10-s + 0.709·11-s − 4.14·12-s + 13-s − 2.20·14-s + 1.49·15-s + 3.90·16-s + 1.77·17-s + 2.41·18-s − 2.77·20-s + 1.70·21-s + 1.37·22-s + 0.241·23-s − 5.14·24-s + 25-s + 1.94·26-s − 0.360·27-s − 3.14·28-s + 2.90·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2015 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2015 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(2.103658311\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.103658311\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 5 | \( 1 + T \) |
| 13 | \( 1 - T \) |
| 31 | \( 1 + T \) |
good | 2 | \( 1 - 1.94T + T^{2} \) |
| 3 | \( 1 + 1.49T + T^{2} \) |
| 7 | \( 1 + 1.13T + T^{2} \) |
| 11 | \( 1 - 0.709T + T^{2} \) |
| 17 | \( 1 - 1.77T + T^{2} \) |
| 19 | \( 1 - T^{2} \) |
| 23 | \( 1 - 0.241T + T^{2} \) |
| 29 | \( 1 - T^{2} \) |
| 37 | \( 1 - T^{2} \) |
| 41 | \( 1 - T^{2} \) |
| 43 | \( 1 + 1.94T + T^{2} \) |
| 47 | \( 1 + 0.241T + T^{2} \) |
| 53 | \( 1 + 0.709T + T^{2} \) |
| 59 | \( 1 - T^{2} \) |
| 61 | \( 1 - T^{2} \) |
| 67 | \( 1 - 0.709T + T^{2} \) |
| 71 | \( 1 - T^{2} \) |
| 73 | \( 1 - T^{2} \) |
| 79 | \( 1 - T^{2} \) |
| 83 | \( 1 - T^{2} \) |
| 89 | \( 1 + 1.13T + T^{2} \) |
| 97 | \( 1 - 1.49T + T^{2} \) |
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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
−9.748095285475432009856871183692, −8.177984672036051286615261573008, −7.11461840075603576610267928701, −6.66615138907150370775517314705, −5.96998792872770954554930417310, −5.38913875767205262704620256587, −4.53252905307959028205511788791, −3.55247825461183670086020724073, −3.30138835769471909622945047789, −1.29692522388745557124190793142,
1.29692522388745557124190793142, 3.30138835769471909622945047789, 3.55247825461183670086020724073, 4.53252905307959028205511788791, 5.38913875767205262704620256587, 5.96998792872770954554930417310, 6.66615138907150370775517314705, 7.11461840075603576610267928701, 8.177984672036051286615261573008, 9.748095285475432009856871183692