| L(s) = 1 | − 1.51i·3-s − 3.49i·7-s + 0.705·9-s − 4.35·11-s − 3.67i·13-s − 5.18i·17-s − 2.08·19-s − 5.29·21-s − i·23-s − 5.61i·27-s + 1.24·29-s + 4.82·31-s + 6.59i·33-s + 1.04i·37-s − 5.57·39-s + ⋯ |
| L(s) = 1 | − 0.874i·3-s − 1.32i·7-s + 0.235·9-s − 1.31·11-s − 1.02i·13-s − 1.25i·17-s − 0.478·19-s − 1.15·21-s − 0.208i·23-s − 1.08i·27-s + 0.231·29-s + 0.866·31-s + 1.14i·33-s + 0.171i·37-s − 0.892·39-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4600 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.894 - 0.447i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4600 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.894 - 0.447i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.219896911\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.219896911\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 \) |
| 23 | \( 1 + iT \) |
| good | 3 | \( 1 + 1.51iT - 3T^{2} \) |
| 7 | \( 1 + 3.49iT - 7T^{2} \) |
| 11 | \( 1 + 4.35T + 11T^{2} \) |
| 13 | \( 1 + 3.67iT - 13T^{2} \) |
| 17 | \( 1 + 5.18iT - 17T^{2} \) |
| 19 | \( 1 + 2.08T + 19T^{2} \) |
| 29 | \( 1 - 1.24T + 29T^{2} \) |
| 31 | \( 1 - 4.82T + 31T^{2} \) |
| 37 | \( 1 - 1.04iT - 37T^{2} \) |
| 41 | \( 1 - 9.05T + 41T^{2} \) |
| 43 | \( 1 + 10.4iT - 43T^{2} \) |
| 47 | \( 1 - 12.8iT - 47T^{2} \) |
| 53 | \( 1 + 9.37iT - 53T^{2} \) |
| 59 | \( 1 + 14.1T + 59T^{2} \) |
| 61 | \( 1 + 9.29T + 61T^{2} \) |
| 67 | \( 1 - 4.44iT - 67T^{2} \) |
| 71 | \( 1 - 2.45T + 71T^{2} \) |
| 73 | \( 1 - 13.2iT - 73T^{2} \) |
| 79 | \( 1 + 7.72T + 79T^{2} \) |
| 83 | \( 1 - 2.97iT - 83T^{2} \) |
| 89 | \( 1 - 14.6T + 89T^{2} \) |
| 97 | \( 1 - 9.37iT - 97T^{2} \) |
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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
−7.73201505201349950772454917271, −7.34237313593921467291790327980, −6.65036452936328651446533266864, −5.79952926245925200912060659592, −4.87267211445758597899080277480, −4.24814600387404678214425924138, −3.07430699601661760468983898501, −2.39242332520993070101829472065, −1.08359982691845161745317578006, −0.36218022592829274741426318712,
1.69769897815204959121875810364, 2.53193034757797898648045646507, 3.41850314489998900156275896760, 4.48617787566647488078274585976, 4.82186504598412069458465508208, 5.89632786819799783728237079254, 6.23850359318892174924093686617, 7.43564570402269083357924281383, 8.115139790156896431114451605880, 8.904349727963857758430500677541
