| L(s) = 1 | − 0.267·5-s − 0.464·13-s − 2.26·17-s − 4.92·25-s − 6.66·29-s + 11.3·37-s − 8·41-s − 7·49-s − 4·53-s − 5.39·61-s + 0.124·65-s + 10.8·73-s + 0.607·85-s − 16.6·89-s − 18·97-s − 20·101-s + 20.3·109-s − 20.1·113-s + ⋯ |
| L(s) = 1 | − 0.119·5-s − 0.128·13-s − 0.550·17-s − 0.985·25-s − 1.23·29-s + 1.87·37-s − 1.24·41-s − 49-s − 0.549·53-s − 0.690·61-s + 0.0154·65-s + 1.27·73-s + 0.0659·85-s − 1.76·89-s − 1.82·97-s − 1.99·101-s + 1.94·109-s − 1.89·113-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2592 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2592 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 3 | \( 1 \) |
| good | 5 | \( 1 + 0.267T + 5T^{2} \) |
| 7 | \( 1 + 7T^{2} \) |
| 11 | \( 1 + 11T^{2} \) |
| 13 | \( 1 + 0.464T + 13T^{2} \) |
| 17 | \( 1 + 2.26T + 17T^{2} \) |
| 19 | \( 1 + 19T^{2} \) |
| 23 | \( 1 + 23T^{2} \) |
| 29 | \( 1 + 6.66T + 29T^{2} \) |
| 31 | \( 1 + 31T^{2} \) |
| 37 | \( 1 - 11.3T + 37T^{2} \) |
| 41 | \( 1 + 8T + 41T^{2} \) |
| 43 | \( 1 + 43T^{2} \) |
| 47 | \( 1 + 47T^{2} \) |
| 53 | \( 1 + 4T + 53T^{2} \) |
| 59 | \( 1 + 59T^{2} \) |
| 61 | \( 1 + 5.39T + 61T^{2} \) |
| 67 | \( 1 + 67T^{2} \) |
| 71 | \( 1 + 71T^{2} \) |
| 73 | \( 1 - 10.8T + 73T^{2} \) |
| 79 | \( 1 + 79T^{2} \) |
| 83 | \( 1 + 83T^{2} \) |
| 89 | \( 1 + 16.6T + 89T^{2} \) |
| 97 | \( 1 + 18T + 97T^{2} \) |
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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
−8.379859570239159023629443917060, −7.82336450262115715202956535392, −6.98266470829251783263143876777, −6.19772515397853326248217132886, −5.40040882629185586874703660705, −4.47241613502914019846643175068, −3.69624076068292913409317372452, −2.63528933529652172863687198528, −1.58953134256885053149055926902, 0,
1.58953134256885053149055926902, 2.63528933529652172863687198528, 3.69624076068292913409317372452, 4.47241613502914019846643175068, 5.40040882629185586874703660705, 6.19772515397853326248217132886, 6.98266470829251783263143876777, 7.82336450262115715202956535392, 8.379859570239159023629443917060
