| L(s) = 1 | − 2-s − 3-s + 4-s + 6-s − 2·7-s − 8-s + 9-s − 0.470·11-s − 12-s − 6.47·13-s + 2·14-s + 16-s − 7.04·17-s − 18-s − 7.04·19-s + 2·21-s + 0.470·22-s − 6.94·23-s + 24-s + 6.47·26-s − 27-s − 2·28-s − 6.94·29-s − 31-s − 32-s + 0.470·33-s + 7.04·34-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 0.577·3-s + 0.5·4-s + 0.408·6-s − 0.755·7-s − 0.353·8-s + 0.333·9-s − 0.141·11-s − 0.288·12-s − 1.79·13-s + 0.534·14-s + 0.250·16-s − 1.70·17-s − 0.235·18-s − 1.61·19-s + 0.436·21-s + 0.100·22-s − 1.44·23-s + 0.204·24-s + 1.26·26-s − 0.192·27-s − 0.377·28-s − 1.28·29-s − 0.179·31-s − 0.176·32-s + 0.0819·33-s + 1.20·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4650 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4650 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.1134277674\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.1134277674\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) |
| 31 | \( 1 + T \) |
| good | 7 | \( 1 + 2T + 7T^{2} \) |
| 11 | \( 1 + 0.470T + 11T^{2} \) |
| 13 | \( 1 + 6.47T + 13T^{2} \) |
| 17 | \( 1 + 7.04T + 17T^{2} \) |
| 19 | \( 1 + 7.04T + 19T^{2} \) |
| 23 | \( 1 + 6.94T + 23T^{2} \) |
| 29 | \( 1 + 6.94T + 29T^{2} \) |
| 37 | \( 1 - 1.78T + 37T^{2} \) |
| 41 | \( 1 - 2T + 41T^{2} \) |
| 43 | \( 1 - 0.210T + 43T^{2} \) |
| 47 | \( 1 - 7.04T + 47T^{2} \) |
| 53 | \( 1 - 3.15T + 53T^{2} \) |
| 59 | \( 1 + 7.15T + 59T^{2} \) |
| 61 | \( 1 - 11.2T + 61T^{2} \) |
| 67 | \( 1 - 8.26T + 67T^{2} \) |
| 71 | \( 1 - 0.260T + 71T^{2} \) |
| 73 | \( 1 + 11.8T + 73T^{2} \) |
| 79 | \( 1 + 1.89T + 79T^{2} \) |
| 83 | \( 1 - 11.7T + 83T^{2} \) |
| 89 | \( 1 - 12.0T + 89T^{2} \) |
| 97 | \( 1 - 3.52T + 97T^{2} \) |
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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
−8.296798131607961280086353994881, −7.54279158825487356898408327384, −6.83359485348203258374259818153, −6.34483667576828508476832043473, −5.54611696313498187488460700140, −4.55487493926639681205846655531, −3.89153613793978506193382659687, −2.44900472588570954510904864135, −2.06610244894528914292120562700, −0.19583818587752804451851287562,
0.19583818587752804451851287562, 2.06610244894528914292120562700, 2.44900472588570954510904864135, 3.89153613793978506193382659687, 4.55487493926639681205846655531, 5.54611696313498187488460700140, 6.34483667576828508476832043473, 6.83359485348203258374259818153, 7.54279158825487356898408327384, 8.296798131607961280086353994881
