| L(s) = 1 | + 2-s + 3-s + 4-s + 6-s − 1.70·7-s + 8-s + 9-s − 1.70·11-s + 12-s − 13-s − 1.70·14-s + 16-s + 5.70·17-s + 18-s + 4.70·19-s − 1.70·21-s − 1.70·22-s + 24-s − 26-s + 27-s − 1.70·28-s + 6.40·29-s + 9.10·31-s + 32-s − 1.70·33-s + 5.70·34-s + 36-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 0.577·3-s + 0.5·4-s + 0.408·6-s − 0.643·7-s + 0.353·8-s + 0.333·9-s − 0.513·11-s + 0.288·12-s − 0.277·13-s − 0.454·14-s + 0.250·16-s + 1.38·17-s + 0.235·18-s + 1.07·19-s − 0.371·21-s − 0.362·22-s + 0.204·24-s − 0.196·26-s + 0.192·27-s − 0.321·28-s + 1.18·29-s + 1.63·31-s + 0.176·32-s − 0.296·33-s + 0.977·34-s + 0.166·36-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1950 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1950 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.336701491\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.336701491\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) |
| 13 | \( 1 + T \) |
| good | 7 | \( 1 + 1.70T + 7T^{2} \) |
| 11 | \( 1 + 1.70T + 11T^{2} \) |
| 17 | \( 1 - 5.70T + 17T^{2} \) |
| 19 | \( 1 - 4.70T + 19T^{2} \) |
| 23 | \( 1 + 23T^{2} \) |
| 29 | \( 1 - 6.40T + 29T^{2} \) |
| 31 | \( 1 - 9.10T + 31T^{2} \) |
| 37 | \( 1 - 4.70T + 37T^{2} \) |
| 41 | \( 1 - 2.70T + 41T^{2} \) |
| 43 | \( 1 - 1.40T + 43T^{2} \) |
| 47 | \( 1 - 7T + 47T^{2} \) |
| 53 | \( 1 + 10.4T + 53T^{2} \) |
| 59 | \( 1 + 3.70T + 59T^{2} \) |
| 61 | \( 1 + 5.10T + 61T^{2} \) |
| 67 | \( 1 - 6.40T + 67T^{2} \) |
| 71 | \( 1 - 4.70T + 71T^{2} \) |
| 73 | \( 1 + 12T + 73T^{2} \) |
| 79 | \( 1 - 0.701T + 79T^{2} \) |
| 83 | \( 1 + 4.29T + 83T^{2} \) |
| 89 | \( 1 + 1.40T + 89T^{2} \) |
| 97 | \( 1 + 15.4T + 97T^{2} \) |
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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
−9.366598744638533258247985214991, −8.131178516926033935844877689244, −7.70040586276990340784054506407, −6.76698181755503116493429033525, −5.95282924395101628605757628677, −5.09540059657061842625894590634, −4.21705929344026704736219487744, −3.09192715743906049803963333835, −2.75128331148739308030647300736, −1.15409195685874792732061272272,
1.15409195685874792732061272272, 2.75128331148739308030647300736, 3.09192715743906049803963333835, 4.21705929344026704736219487744, 5.09540059657061842625894590634, 5.95282924395101628605757628677, 6.76698181755503116493429033525, 7.70040586276990340784054506407, 8.131178516926033935844877689244, 9.366598744638533258247985214991
