| L(s) = 1 | − 0.540·3-s + 1.15·7-s − 2.70·9-s + 2.10·11-s + 5.59·13-s + 0.244·17-s + 1.45·19-s − 0.625·21-s − 23-s + 3.08·27-s + 4.29·29-s + 3.19·31-s − 1.13·33-s − 0.807·37-s − 3.02·39-s − 1.59·41-s − 5.98·43-s − 0.624·47-s − 5.66·49-s − 0.132·51-s − 0.536·53-s − 0.784·57-s − 1.03·59-s + 3.77·61-s − 3.13·63-s − 4.61·67-s + 0.540·69-s + ⋯ |
| L(s) = 1 | − 0.312·3-s + 0.437·7-s − 0.902·9-s + 0.635·11-s + 1.55·13-s + 0.0592·17-s + 0.332·19-s − 0.136·21-s − 0.208·23-s + 0.593·27-s + 0.797·29-s + 0.574·31-s − 0.198·33-s − 0.132·37-s − 0.484·39-s − 0.249·41-s − 0.912·43-s − 0.0911·47-s − 0.808·49-s − 0.0184·51-s − 0.0736·53-s − 0.103·57-s − 0.135·59-s + 0.483·61-s − 0.394·63-s − 0.563·67-s + 0.0650·69-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4600 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4600 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.903051989\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.903051989\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + T \) |
| good | 3 | \( 1 + 0.540T + 3T^{2} \) |
| 7 | \( 1 - 1.15T + 7T^{2} \) |
| 11 | \( 1 - 2.10T + 11T^{2} \) |
| 13 | \( 1 - 5.59T + 13T^{2} \) |
| 17 | \( 1 - 0.244T + 17T^{2} \) |
| 19 | \( 1 - 1.45T + 19T^{2} \) |
| 29 | \( 1 - 4.29T + 29T^{2} \) |
| 31 | \( 1 - 3.19T + 31T^{2} \) |
| 37 | \( 1 + 0.807T + 37T^{2} \) |
| 41 | \( 1 + 1.59T + 41T^{2} \) |
| 43 | \( 1 + 5.98T + 43T^{2} \) |
| 47 | \( 1 + 0.624T + 47T^{2} \) |
| 53 | \( 1 + 0.536T + 53T^{2} \) |
| 59 | \( 1 + 1.03T + 59T^{2} \) |
| 61 | \( 1 - 3.77T + 61T^{2} \) |
| 67 | \( 1 + 4.61T + 67T^{2} \) |
| 71 | \( 1 - 6.77T + 71T^{2} \) |
| 73 | \( 1 + 8.87T + 73T^{2} \) |
| 79 | \( 1 - 14.8T + 79T^{2} \) |
| 83 | \( 1 - 6.11T + 83T^{2} \) |
| 89 | \( 1 - 5.79T + 89T^{2} \) |
| 97 | \( 1 + 2.38T + 97T^{2} \) |
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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
−8.424337626284415976084443883641, −7.74211823823229297304291162278, −6.57476329506424513891783557102, −6.26497252948264387493915638052, −5.41497952869519375791251420181, −4.67917758134640289061137481508, −3.71323422112012808415108459046, −3.02874886725870270893449927919, −1.78199110896544196441105848703, −0.814415899322667552186692860861,
0.814415899322667552186692860861, 1.78199110896544196441105848703, 3.02874886725870270893449927919, 3.71323422112012808415108459046, 4.67917758134640289061137481508, 5.41497952869519375791251420181, 6.26497252948264387493915638052, 6.57476329506424513891783557102, 7.74211823823229297304291162278, 8.424337626284415976084443883641
