L(s) = 1 | + 1.77·2-s + 0.241·3-s + 2.13·4-s + 5-s + 0.426·6-s − 0.709·7-s + 2.01·8-s − 0.941·9-s + 1.77·10-s − 1.49·11-s + 0.514·12-s + 13-s − 1.25·14-s + 0.241·15-s + 1.42·16-s + 1.13·17-s − 1.66·18-s + 2.13·20-s − 0.170·21-s − 2.65·22-s − 1.94·23-s + 0.485·24-s + 25-s + 1.77·26-s − 0.468·27-s − 1.51·28-s + 0.426·30-s + ⋯ |
L(s) = 1 | + 1.77·2-s + 0.241·3-s + 2.13·4-s + 5-s + 0.426·6-s − 0.709·7-s + 2.01·8-s − 0.941·9-s + 1.77·10-s − 1.49·11-s + 0.514·12-s + 13-s − 1.25·14-s + 0.241·15-s + 1.42·16-s + 1.13·17-s − 1.66·18-s + 2.13·20-s − 0.170·21-s − 2.65·22-s − 1.94·23-s + 0.485·24-s + 25-s + 1.77·26-s − 0.468·27-s − 1.51·28-s + 0.426·30-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
\(L(\frac{1}{2})\) |
\(\approx\) |
\(3.443829511\) |
\(L(\frac12)\) |
\(\approx\) |
\(3.443829511\) |
\(L(1)\) |
|
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 | \( 1 - 1.77T + T^{2} \) |
| 3 | \( 1 - 0.241T + T^{2} \) |
| 7 | \( 1 + 0.709T + T^{2} \) |
| 11 | \( 1 + 1.49T + T^{2} \) |
| 17 | \( 1 - 1.13T + T^{2} \) |
| 19 | \( 1 - T^{2} \) |
| 23 | \( 1 + 1.94T + T^{2} \) |
| 29 | \( 1 - T^{2} \) |
| 37 | \( 1 - T^{2} \) |
| 41 | \( 1 - T^{2} \) |
| 43 | \( 1 - 1.77T + T^{2} \) |
| 47 | \( 1 + 1.94T + T^{2} \) |
| 53 | \( 1 + 1.49T + T^{2} \) |
| 59 | \( 1 - T^{2} \) |
| 61 | \( 1 - T^{2} \) |
| 67 | \( 1 + 1.49T + T^{2} \) |
| 71 | \( 1 - T^{2} \) |
| 73 | \( 1 - T^{2} \) |
| 79 | \( 1 - T^{2} \) |
| 83 | \( 1 - T^{2} \) |
| 89 | \( 1 + 0.709T + T^{2} \) |
| 97 | \( 1 - 0.241T + T^{2} \) |
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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
−9.553910796934658229439010208358, −8.331853722329919319195795773162, −7.67714064493939445329051346477, −6.37577170980708743920650432453, −5.97115563268794931281812975274, −5.49584272938350748614156779193, −4.54409887523622501690404521631, −3.31776820361824830169502380280, −2.91608484996336079661788366987, −1.94733314344904275299982961783,
1.94733314344904275299982961783, 2.91608484996336079661788366987, 3.31776820361824830169502380280, 4.54409887523622501690404521631, 5.49584272938350748614156779193, 5.97115563268794931281812975274, 6.37577170980708743920650432453, 7.67714064493939445329051346477, 8.331853722329919319195795773162, 9.553910796934658229439010208358