L(s) = 1 | − 2-s − 3-s + 4-s + 1.23·5-s + 6-s + 2.61·7-s − 8-s + 9-s − 1.23·10-s + 4·11-s − 12-s − 4·13-s − 2.61·14-s − 1.23·15-s + 16-s − 2.85·17-s − 18-s + 1.14·19-s + 1.23·20-s − 2.61·21-s − 4·22-s + 4.61·23-s + 24-s − 3.47·25-s + 4·26-s − 27-s + 2.61·28-s + ⋯ |
L(s) = 1 | − 0.707·2-s − 0.577·3-s + 0.5·4-s + 0.552·5-s + 0.408·6-s + 0.989·7-s − 0.353·8-s + 0.333·9-s − 0.390·10-s + 1.20·11-s − 0.288·12-s − 1.10·13-s − 0.699·14-s − 0.319·15-s + 0.250·16-s − 0.692·17-s − 0.235·18-s + 0.262·19-s + 0.276·20-s − 0.571·21-s − 0.852·22-s + 0.962·23-s + 0.204·24-s − 0.694·25-s + 0.784·26-s − 0.192·27-s + 0.494·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6078 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6078 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.493575679\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.493575679\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 + T \) |
| 1013 | \( 1 - T \) |
good | 5 | \( 1 - 1.23T + 5T^{2} \) |
| 7 | \( 1 - 2.61T + 7T^{2} \) |
| 11 | \( 1 - 4T + 11T^{2} \) |
| 13 | \( 1 + 4T + 13T^{2} \) |
| 17 | \( 1 + 2.85T + 17T^{2} \) |
| 19 | \( 1 - 1.14T + 19T^{2} \) |
| 23 | \( 1 - 4.61T + 23T^{2} \) |
| 29 | \( 1 + 6T + 29T^{2} \) |
| 31 | \( 1 - 7.32T + 31T^{2} \) |
| 37 | \( 1 - 3.61T + 37T^{2} \) |
| 41 | \( 1 + 1.09T + 41T^{2} \) |
| 43 | \( 1 + 3.09T + 43T^{2} \) |
| 47 | \( 1 - 10T + 47T^{2} \) |
| 53 | \( 1 + 11.3T + 53T^{2} \) |
| 59 | \( 1 - 4.14T + 59T^{2} \) |
| 61 | \( 1 + 11.6T + 61T^{2} \) |
| 67 | \( 1 - 10.9T + 67T^{2} \) |
| 71 | \( 1 - 5.85T + 71T^{2} \) |
| 73 | \( 1 - 10.5T + 73T^{2} \) |
| 79 | \( 1 - 8.94T + 79T^{2} \) |
| 83 | \( 1 - 9.70T + 83T^{2} \) |
| 89 | \( 1 - 11.7T + 89T^{2} \) |
| 97 | \( 1 - 8.85T + 97T^{2} \) |
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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
−7.948933351871451764592169890768, −7.46648728653352703499661086464, −6.60684164966449480517289377867, −6.16896599446235010377929303476, −5.14010098620764762232183156530, −4.68981011648269308524935635746, −3.66271837216641014801264686148, −2.38140560357954548243718590884, −1.71704270101475389908264218978, −0.76139744220237833636670332256,
0.76139744220237833636670332256, 1.71704270101475389908264218978, 2.38140560357954548243718590884, 3.66271837216641014801264686148, 4.68981011648269308524935635746, 5.14010098620764762232183156530, 6.16896599446235010377929303476, 6.60684164966449480517289377867, 7.46648728653352703499661086464, 7.948933351871451764592169890768