| L(s) = 1 | + 2-s + 3-s + 4-s + 6-s − 4.42·7-s + 8-s + 9-s − 5.80·11-s + 12-s + 6.42·13-s − 4.42·14-s + 16-s + 3.37·17-s + 18-s − 19-s − 4.42·21-s − 5.80·22-s + 6.42·23-s + 24-s + 6.42·26-s + 27-s − 4.42·28-s + 7.80·29-s + 9.05·31-s + 32-s − 5.80·33-s + 3.37·34-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 0.577·3-s + 0.5·4-s + 0.408·6-s − 1.67·7-s + 0.353·8-s + 0.333·9-s − 1.75·11-s + 0.288·12-s + 1.78·13-s − 1.18·14-s + 0.250·16-s + 0.819·17-s + 0.235·18-s − 0.229·19-s − 0.966·21-s − 1.23·22-s + 1.34·23-s + 0.204·24-s + 1.26·26-s + 0.192·27-s − 0.836·28-s + 1.44·29-s + 1.62·31-s + 0.176·32-s − 1.01·33-s + 0.579·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2850 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2850 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.067505923\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.067505923\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) |
| 19 | \( 1 + T \) |
| good | 7 | \( 1 + 4.42T + 7T^{2} \) |
| 11 | \( 1 + 5.80T + 11T^{2} \) |
| 13 | \( 1 - 6.42T + 13T^{2} \) |
| 17 | \( 1 - 3.37T + 17T^{2} \) |
| 23 | \( 1 - 6.42T + 23T^{2} \) |
| 29 | \( 1 - 7.80T + 29T^{2} \) |
| 31 | \( 1 - 9.05T + 31T^{2} \) |
| 37 | \( 1 + 3.67T + 37T^{2} \) |
| 41 | \( 1 - 4.42T + 41T^{2} \) |
| 43 | \( 1 - 1.05T + 43T^{2} \) |
| 47 | \( 1 + 5.18T + 47T^{2} \) |
| 53 | \( 1 - 4.75T + 53T^{2} \) |
| 59 | \( 1 - 4.62T + 59T^{2} \) |
| 61 | \( 1 - 2T + 61T^{2} \) |
| 67 | \( 1 + 2.75T + 67T^{2} \) |
| 71 | \( 1 - 7.61T + 71T^{2} \) |
| 73 | \( 1 + 11.6T + 73T^{2} \) |
| 79 | \( 1 - 2.94T + 79T^{2} \) |
| 83 | \( 1 - 0.133T + 83T^{2} \) |
| 89 | \( 1 + 3.18T + 89T^{2} \) |
| 97 | \( 1 - 11.4T + 97T^{2} \) |
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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
−8.618856837152534003301562045052, −8.109693810052678754520084827256, −7.10601437125796402543833101237, −6.43045289009265658999750085701, −5.78133640412975049654380276877, −4.87725343105415445933195318676, −3.80000799176177355598201842257, −3.04491240443393951626008519290, −2.67580606341784920537303206430, −0.959314908362890965376886831490,
0.959314908362890965376886831490, 2.67580606341784920537303206430, 3.04491240443393951626008519290, 3.80000799176177355598201842257, 4.87725343105415445933195318676, 5.78133640412975049654380276877, 6.43045289009265658999750085701, 7.10601437125796402543833101237, 8.109693810052678754520084827256, 8.618856837152534003301562045052
