| L(s) = 1 | − 2·3-s − 5-s + 4·7-s + 9-s + 6·13-s + 2·15-s − 8·21-s − 8·23-s + 25-s + 4·27-s − 8·31-s − 4·35-s − 12·39-s − 6·41-s + 2·43-s − 45-s + 2·47-s + 9·49-s − 6·53-s + 12·59-s + 6·61-s + 4·63-s − 6·65-s + 16·67-s + 16·69-s − 4·73-s − 2·75-s + ⋯ |
| L(s) = 1 | − 1.15·3-s − 0.447·5-s + 1.51·7-s + 1/3·9-s + 1.66·13-s + 0.516·15-s − 1.74·21-s − 1.66·23-s + 1/5·25-s + 0.769·27-s − 1.43·31-s − 0.676·35-s − 1.92·39-s − 0.937·41-s + 0.304·43-s − 0.149·45-s + 0.291·47-s + 9/7·49-s − 0.824·53-s + 1.56·59-s + 0.768·61-s + 0.503·63-s − 0.744·65-s + 1.95·67-s + 1.92·69-s − 0.468·73-s − 0.230·75-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 33640 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 33640 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 + T \) |
| 29 | \( 1 \) |
| good | 3 | \( 1 + 2 T + p T^{2} \) |
| 7 | \( 1 - 4 T + p T^{2} \) |
| 11 | \( 1 + p T^{2} \) |
| 13 | \( 1 - 6 T + p T^{2} \) |
| 17 | \( 1 + p T^{2} \) |
| 19 | \( 1 + p T^{2} \) |
| 23 | \( 1 + 8 T + p T^{2} \) |
| 31 | \( 1 + 8 T + p T^{2} \) |
| 37 | \( 1 + p T^{2} \) |
| 41 | \( 1 + 6 T + p T^{2} \) |
| 43 | \( 1 - 2 T + p T^{2} \) |
| 47 | \( 1 - 2 T + p T^{2} \) |
| 53 | \( 1 + 6 T + p T^{2} \) |
| 59 | \( 1 - 12 T + p T^{2} \) |
| 61 | \( 1 - 6 T + p T^{2} \) |
| 67 | \( 1 - 16 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 + 4 T + p T^{2} \) |
| 79 | \( 1 - 8 T + p T^{2} \) |
| 83 | \( 1 - 8 T + p T^{2} \) |
| 89 | \( 1 - 10 T + p T^{2} \) |
| 97 | \( 1 + 8 T + p T^{2} \) |
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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
−15.35051506826745, −14.62618005807790, −14.33192352203785, −13.68737042188838, −13.15683496215417, −12.37269320751578, −11.97150569862655, −11.44592493602655, −11.10778485851403, −10.75326286654639, −10.20125694610470, −9.315279444459757, −8.638656978218095, −8.081201721202322, −7.908275999801334, −6.874269638037504, −6.481392165467345, −5.640058424764240, −5.419784636220776, −4.758370699980516, −3.878028809859158, −3.750221313133976, −2.405396714466148, −1.620508357358439, −0.9990839132611076, 0,
0.9990839132611076, 1.620508357358439, 2.405396714466148, 3.750221313133976, 3.878028809859158, 4.758370699980516, 5.419784636220776, 5.640058424764240, 6.481392165467345, 6.874269638037504, 7.908275999801334, 8.081201721202322, 8.638656978218095, 9.315279444459757, 10.20125694610470, 10.75326286654639, 11.10778485851403, 11.44592493602655, 11.97150569862655, 12.37269320751578, 13.15683496215417, 13.68737042188838, 14.33192352203785, 14.62618005807790, 15.35051506826745
