| L(s) = 1 | − 2·2-s + 3·3-s + 4·4-s − 5·5-s − 6·6-s + 34.7·7-s − 8·8-s + 9·9-s + 10·10-s + 35.0·11-s + 12·12-s + 87.2·13-s − 69.5·14-s − 15·15-s + 16·16-s + 69.2·17-s − 18·18-s − 63.9·19-s − 20·20-s + 104.·21-s − 70.0·22-s + 205.·23-s − 24·24-s + 25·25-s − 174.·26-s + 27·27-s + 139.·28-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 0.577·3-s + 0.5·4-s − 0.447·5-s − 0.408·6-s + 1.87·7-s − 0.353·8-s + 0.333·9-s + 0.316·10-s + 0.959·11-s + 0.288·12-s + 1.86·13-s − 1.32·14-s − 0.258·15-s + 0.250·16-s + 0.988·17-s − 0.235·18-s − 0.771·19-s − 0.223·20-s + 1.08·21-s − 0.678·22-s + 1.86·23-s − 0.204·24-s + 0.200·25-s − 1.31·26-s + 0.192·27-s + 0.939·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 930 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 930 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(2.768585277\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.768585277\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + 2T \) |
| 3 | \( 1 - 3T \) |
| 5 | \( 1 + 5T \) |
| 31 | \( 1 - 31T \) |
| good | 7 | \( 1 - 34.7T + 343T^{2} \) |
| 11 | \( 1 - 35.0T + 1.33e3T^{2} \) |
| 13 | \( 1 - 87.2T + 2.19e3T^{2} \) |
| 17 | \( 1 - 69.2T + 4.91e3T^{2} \) |
| 19 | \( 1 + 63.9T + 6.85e3T^{2} \) |
| 23 | \( 1 - 205.T + 1.21e4T^{2} \) |
| 29 | \( 1 + 194.T + 2.43e4T^{2} \) |
| 37 | \( 1 - 9.77T + 5.06e4T^{2} \) |
| 41 | \( 1 + 373.T + 6.89e4T^{2} \) |
| 43 | \( 1 - 181.T + 7.95e4T^{2} \) |
| 47 | \( 1 + 380.T + 1.03e5T^{2} \) |
| 53 | \( 1 + 160.T + 1.48e5T^{2} \) |
| 59 | \( 1 + 722.T + 2.05e5T^{2} \) |
| 61 | \( 1 + 401.T + 2.26e5T^{2} \) |
| 67 | \( 1 - 1.05e3T + 3.00e5T^{2} \) |
| 71 | \( 1 - 331.T + 3.57e5T^{2} \) |
| 73 | \( 1 + 198.T + 3.89e5T^{2} \) |
| 79 | \( 1 + 583.T + 4.93e5T^{2} \) |
| 83 | \( 1 + 511.T + 5.71e5T^{2} \) |
| 89 | \( 1 - 269.T + 7.04e5T^{2} \) |
| 97 | \( 1 + 793.T + 9.12e5T^{2} \) |
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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
−9.389574861079288616655283095828, −8.638593300674309823857143957238, −8.243640046640153039534980089087, −7.45908734782716910507656960079, −6.49517050740824131794573456036, −5.28436796702164307428174890751, −4.17483644066731800415649909787, −3.27714878557570186341818480965, −1.65066137345301480658309408600, −1.16178514869251061743875810970,
1.16178514869251061743875810970, 1.65066137345301480658309408600, 3.27714878557570186341818480965, 4.17483644066731800415649909787, 5.28436796702164307428174890751, 6.49517050740824131794573456036, 7.45908734782716910507656960079, 8.243640046640153039534980089087, 8.638593300674309823857143957238, 9.389574861079288616655283095828
