| L(s) = 1 | + 3.04·3-s + 6.24·9-s + 4.37·11-s + 0.657·13-s + 5.11·17-s − 5.03·19-s + 0.0521·23-s + 9.88·27-s + 9.61·29-s + 2.59·31-s + 13.2·33-s + 9.47·37-s + 2·39-s − 1.72·41-s − 11.4·43-s − 10.3·47-s + 15.5·51-s + 11.6·53-s − 15.3·57-s + 0.682·59-s + 2.24·61-s + 3.60·67-s + 0.158·69-s + 1.01·71-s − 10.0·73-s − 15.1·79-s + 11.3·81-s + ⋯ |
| L(s) = 1 | + 1.75·3-s + 2.08·9-s + 1.31·11-s + 0.182·13-s + 1.24·17-s − 1.15·19-s + 0.0108·23-s + 1.90·27-s + 1.78·29-s + 0.466·31-s + 2.31·33-s + 1.55·37-s + 0.320·39-s − 0.269·41-s − 1.75·43-s − 1.50·47-s + 2.17·51-s + 1.60·53-s − 2.02·57-s + 0.0889·59-s + 0.287·61-s + 0.440·67-s + 0.0190·69-s + 0.120·71-s − 1.17·73-s − 1.70·79-s + 1.25·81-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 9800 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 9800 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(5.204832393\) |
| \(L(\frac12)\) |
\(\approx\) |
\(5.204832393\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 \) |
| 7 | \( 1 \) |
| good | 3 | \( 1 - 3.04T + 3T^{2} \) |
| 11 | \( 1 - 4.37T + 11T^{2} \) |
| 13 | \( 1 - 0.657T + 13T^{2} \) |
| 17 | \( 1 - 5.11T + 17T^{2} \) |
| 19 | \( 1 + 5.03T + 19T^{2} \) |
| 23 | \( 1 - 0.0521T + 23T^{2} \) |
| 29 | \( 1 - 9.61T + 29T^{2} \) |
| 31 | \( 1 - 2.59T + 31T^{2} \) |
| 37 | \( 1 - 9.47T + 37T^{2} \) |
| 41 | \( 1 + 1.72T + 41T^{2} \) |
| 43 | \( 1 + 11.4T + 43T^{2} \) |
| 47 | \( 1 + 10.3T + 47T^{2} \) |
| 53 | \( 1 - 11.6T + 53T^{2} \) |
| 59 | \( 1 - 0.682T + 59T^{2} \) |
| 61 | \( 1 - 2.24T + 61T^{2} \) |
| 67 | \( 1 - 3.60T + 67T^{2} \) |
| 71 | \( 1 - 1.01T + 71T^{2} \) |
| 73 | \( 1 + 10.0T + 73T^{2} \) |
| 79 | \( 1 + 15.1T + 79T^{2} \) |
| 83 | \( 1 + 9.04T + 83T^{2} \) |
| 89 | \( 1 + 15.3T + 89T^{2} \) |
| 97 | \( 1 + 0.404T + 97T^{2} \) |
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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
−7.994270696334566051543507661879, −6.92899005150035200803368988813, −6.66005110771361988792847656686, −5.69507195633068635147553560700, −4.48016824059510908595483611752, −4.14036786458816861284876680028, −3.25888538530628066401718581523, −2.78920381641755308698610776650, −1.77932756151497985750526014726, −1.09544087864346517638391291206,
1.09544087864346517638391291206, 1.77932756151497985750526014726, 2.78920381641755308698610776650, 3.25888538530628066401718581523, 4.14036786458816861284876680028, 4.48016824059510908595483611752, 5.69507195633068635147553560700, 6.66005110771361988792847656686, 6.92899005150035200803368988813, 7.994270696334566051543507661879
