| L(s) = 1 | + i·3-s − 3.78i·7-s − 9-s − 0.807·11-s + 4.74i·13-s + 1.14i·17-s − 0.0150·19-s + 3.78·21-s − 6.26i·23-s − i·27-s − 3.70·29-s − 1.58·31-s − 0.807i·33-s + 8.54i·37-s − 4.74·39-s + ⋯ |
| L(s) = 1 | + 0.577i·3-s − 1.43i·7-s − 0.333·9-s − 0.243·11-s + 1.31i·13-s + 0.278i·17-s − 0.00344·19-s + 0.826·21-s − 1.30i·23-s − 0.192i·27-s − 0.687·29-s − 0.284·31-s − 0.140i·33-s + 1.40i·37-s − 0.760·39-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 7500 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -i\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7500 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -i\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.312453629\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.312453629\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 - iT \) |
| 5 | \( 1 \) |
| good | 7 | \( 1 + 3.78iT - 7T^{2} \) |
| 11 | \( 1 + 0.807T + 11T^{2} \) |
| 13 | \( 1 - 4.74iT - 13T^{2} \) |
| 17 | \( 1 - 1.14iT - 17T^{2} \) |
| 19 | \( 1 + 0.0150T + 19T^{2} \) |
| 23 | \( 1 + 6.26iT - 23T^{2} \) |
| 29 | \( 1 + 3.70T + 29T^{2} \) |
| 31 | \( 1 + 1.58T + 31T^{2} \) |
| 37 | \( 1 - 8.54iT - 37T^{2} \) |
| 41 | \( 1 - 11.4T + 41T^{2} \) |
| 43 | \( 1 - 10.2iT - 43T^{2} \) |
| 47 | \( 1 - 0.526iT - 47T^{2} \) |
| 53 | \( 1 + 2.94iT - 53T^{2} \) |
| 59 | \( 1 + 11.4T + 59T^{2} \) |
| 61 | \( 1 - 3.14T + 61T^{2} \) |
| 67 | \( 1 + 13.2iT - 67T^{2} \) |
| 71 | \( 1 + 4.91T + 71T^{2} \) |
| 73 | \( 1 + 4.67iT - 73T^{2} \) |
| 79 | \( 1 + 9.27T + 79T^{2} \) |
| 83 | \( 1 - 1.42iT - 83T^{2} \) |
| 89 | \( 1 - 16.1T + 89T^{2} \) |
| 97 | \( 1 + 8.06iT - 97T^{2} \) |
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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
−7.85719050255387516728919637956, −7.53953543992322625873833502087, −6.46975387245232055670671346639, −6.27303393633208097285655760632, −4.92677256815778428840558936693, −4.48147625650836204013676939213, −3.91078758082982642169806735568, −3.07907255638349102293138253110, −2.00632688949364573742853545883, −0.923984526691558106504902575412,
0.36230517406580118319739991843, 1.65322445985135152906390942885, 2.50211688870488661466594102055, 3.08361844910791363913925275925, 4.05104032531818371583805337637, 5.33321072389492747870638729251, 5.58367817071406127762266648493, 6.08183549202147654721879857416, 7.31838543376001124323810439447, 7.54450608492867356654177623363
