| L(s) = 1 | − 0.414·3-s + 5-s − 2.82·9-s − 0.828·11-s + 2·13-s − 0.414·15-s − 7.65·17-s + 5.65·19-s + 5.58·23-s + 25-s + 2.41·27-s − 7.82·29-s − 0.828·31-s + 0.343·33-s + 5.65·37-s − 0.828·39-s + 5.82·41-s + 6.89·43-s − 2.82·45-s − 11.6·47-s + 3.17·51-s − 5.65·53-s − 0.828·55-s − 2.34·57-s + 4·59-s + 6.65·61-s + 2·65-s + ⋯ |
| L(s) = 1 | − 0.239·3-s + 0.447·5-s − 0.942·9-s − 0.249·11-s + 0.554·13-s − 0.106·15-s − 1.85·17-s + 1.29·19-s + 1.16·23-s + 0.200·25-s + 0.464·27-s − 1.45·29-s − 0.148·31-s + 0.0597·33-s + 0.929·37-s − 0.132·39-s + 0.910·41-s + 1.05·43-s − 0.421·45-s − 1.70·47-s + 0.444·51-s − 0.777·53-s − 0.111·55-s − 0.310·57-s + 0.520·59-s + 0.852·61-s + 0.248·65-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3920 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3920 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.584778244\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.584778244\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 - T \) |
| 7 | \( 1 \) |
| good | 3 | \( 1 + 0.414T + 3T^{2} \) |
| 11 | \( 1 + 0.828T + 11T^{2} \) |
| 13 | \( 1 - 2T + 13T^{2} \) |
| 17 | \( 1 + 7.65T + 17T^{2} \) |
| 19 | \( 1 - 5.65T + 19T^{2} \) |
| 23 | \( 1 - 5.58T + 23T^{2} \) |
| 29 | \( 1 + 7.82T + 29T^{2} \) |
| 31 | \( 1 + 0.828T + 31T^{2} \) |
| 37 | \( 1 - 5.65T + 37T^{2} \) |
| 41 | \( 1 - 5.82T + 41T^{2} \) |
| 43 | \( 1 - 6.89T + 43T^{2} \) |
| 47 | \( 1 + 11.6T + 47T^{2} \) |
| 53 | \( 1 + 5.65T + 53T^{2} \) |
| 59 | \( 1 - 4T + 59T^{2} \) |
| 61 | \( 1 - 6.65T + 61T^{2} \) |
| 67 | \( 1 - 12.8T + 67T^{2} \) |
| 71 | \( 1 - 12T + 71T^{2} \) |
| 73 | \( 1 - 3.65T + 73T^{2} \) |
| 79 | \( 1 - 4T + 79T^{2} \) |
| 83 | \( 1 - 4.75T + 83T^{2} \) |
| 89 | \( 1 + 5.34T + 89T^{2} \) |
| 97 | \( 1 - 6T + 97T^{2} \) |
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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
−8.536991822524003986900530973817, −7.76930775890374975677673185224, −6.87217944978441966901243712097, −6.24504635145943290349738884230, −5.46294119801304949711694951755, −4.90532558591845159140230600797, −3.81454527462671571841952861186, −2.88617416048873644755226866481, −2.06481874527269829240606787476, −0.72132031669111113963529318538,
0.72132031669111113963529318538, 2.06481874527269829240606787476, 2.88617416048873644755226866481, 3.81454527462671571841952861186, 4.90532558591845159140230600797, 5.46294119801304949711694951755, 6.24504635145943290349738884230, 6.87217944978441966901243712097, 7.76930775890374975677673185224, 8.536991822524003986900530973817
