L(s) = 1 | + 2-s − 3-s + 4-s − 5-s − 6-s − 1.19·7-s + 8-s + 9-s − 10-s + 3.93·11-s − 12-s − 1.19·14-s + 15-s + 16-s + 1.74·17-s + 18-s + 0.911·19-s − 20-s + 1.19·21-s + 3.93·22-s + 1.24·23-s − 24-s + 25-s − 27-s − 1.19·28-s − 3.47·29-s + 30-s + ⋯ |
L(s) = 1 | + 0.707·2-s − 0.577·3-s + 0.5·4-s − 0.447·5-s − 0.408·6-s − 0.452·7-s + 0.353·8-s + 0.333·9-s − 0.316·10-s + 1.18·11-s − 0.288·12-s − 0.320·14-s + 0.258·15-s + 0.250·16-s + 0.422·17-s + 0.235·18-s + 0.209·19-s − 0.223·20-s + 0.261·21-s + 0.839·22-s + 0.260·23-s − 0.204·24-s + 0.200·25-s − 0.192·27-s − 0.226·28-s − 0.644·29-s + 0.182·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5070 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5070 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.368155493\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.368155493\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 - T \) |
| 3 | \( 1 + T \) |
| 5 | \( 1 + T \) |
| 13 | \( 1 \) |
good | 7 | \( 1 + 1.19T + 7T^{2} \) |
| 11 | \( 1 - 3.93T + 11T^{2} \) |
| 17 | \( 1 - 1.74T + 17T^{2} \) |
| 19 | \( 1 - 0.911T + 19T^{2} \) |
| 23 | \( 1 - 1.24T + 23T^{2} \) |
| 29 | \( 1 + 3.47T + 29T^{2} \) |
| 31 | \( 1 + 2.15T + 31T^{2} \) |
| 37 | \( 1 - 4.80T + 37T^{2} \) |
| 41 | \( 1 - 0.198T + 41T^{2} \) |
| 43 | \( 1 - 3.43T + 43T^{2} \) |
| 47 | \( 1 - 0.902T + 47T^{2} \) |
| 53 | \( 1 + 6.91T + 53T^{2} \) |
| 59 | \( 1 + 10.2T + 59T^{2} \) |
| 61 | \( 1 - 9.92T + 61T^{2} \) |
| 67 | \( 1 + 9.16T + 67T^{2} \) |
| 71 | \( 1 - 11.0T + 71T^{2} \) |
| 73 | \( 1 - 4.13T + 73T^{2} \) |
| 79 | \( 1 - 2.73T + 79T^{2} \) |
| 83 | \( 1 - 15.5T + 83T^{2} \) |
| 89 | \( 1 - 8.57T + 89T^{2} \) |
| 97 | \( 1 - 16.9T + 97T^{2} \) |
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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
−7.961033924298260787709136481310, −7.38665865576330226955795388807, −6.54681505917868944784492901938, −6.16181660529434503061865531025, −5.27933842653781261230825184820, −4.53254758124759930353412735700, −3.76201628531276298401205921058, −3.18236255582664334578096472213, −1.90122093887564684164372944258, −0.792381293472283070253554312995,
0.792381293472283070253554312995, 1.90122093887564684164372944258, 3.18236255582664334578096472213, 3.76201628531276298401205921058, 4.53254758124759930353412735700, 5.27933842653781261230825184820, 6.16181660529434503061865531025, 6.54681505917868944784492901938, 7.38665865576330226955795388807, 7.961033924298260787709136481310