| L(s) = 1 | − 3·3-s + 5·5-s + 20·7-s + 9·9-s − 16·11-s − 58·13-s − 15·15-s + 38·17-s − 4·19-s − 60·21-s − 80·23-s + 25·25-s − 27·27-s − 82·29-s − 8·31-s + 48·33-s + 100·35-s − 426·37-s + 174·39-s − 246·41-s + 524·43-s + 45·45-s − 464·47-s + 57·49-s − 114·51-s + 702·53-s − 80·55-s + ⋯ |
| L(s) = 1 | − 0.577·3-s + 0.447·5-s + 1.07·7-s + 1/3·9-s − 0.438·11-s − 1.23·13-s − 0.258·15-s + 0.542·17-s − 0.0482·19-s − 0.623·21-s − 0.725·23-s + 1/5·25-s − 0.192·27-s − 0.525·29-s − 0.0463·31-s + 0.253·33-s + 0.482·35-s − 1.89·37-s + 0.714·39-s − 0.937·41-s + 1.85·43-s + 0.149·45-s − 1.44·47-s + 0.166·49-s − 0.313·51-s + 1.81·53-s − 0.196·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 960 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 960 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + p T \) |
| 5 | \( 1 - p T \) |
| good | 7 | \( 1 - 20 T + p^{3} T^{2} \) |
| 11 | \( 1 + 16 T + p^{3} T^{2} \) |
| 13 | \( 1 + 58 T + p^{3} T^{2} \) |
| 17 | \( 1 - 38 T + p^{3} T^{2} \) |
| 19 | \( 1 + 4 T + p^{3} T^{2} \) |
| 23 | \( 1 + 80 T + p^{3} T^{2} \) |
| 29 | \( 1 + 82 T + p^{3} T^{2} \) |
| 31 | \( 1 + 8 T + p^{3} T^{2} \) |
| 37 | \( 1 + 426 T + p^{3} T^{2} \) |
| 41 | \( 1 + 6 p T + p^{3} T^{2} \) |
| 43 | \( 1 - 524 T + p^{3} T^{2} \) |
| 47 | \( 1 + 464 T + p^{3} T^{2} \) |
| 53 | \( 1 - 702 T + p^{3} T^{2} \) |
| 59 | \( 1 - 592 T + p^{3} T^{2} \) |
| 61 | \( 1 + 574 T + p^{3} T^{2} \) |
| 67 | \( 1 - 172 T + p^{3} T^{2} \) |
| 71 | \( 1 - 768 T + p^{3} T^{2} \) |
| 73 | \( 1 + 558 T + p^{3} T^{2} \) |
| 79 | \( 1 - 408 T + p^{3} T^{2} \) |
| 83 | \( 1 + 164 T + p^{3} T^{2} \) |
| 89 | \( 1 + 510 T + p^{3} T^{2} \) |
| 97 | \( 1 - 514 T + p^{3} T^{2} \) |
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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
−9.361502774293671511491915199303, −8.269902934722587275039895689800, −7.53992226595484027540629580230, −6.69533610932342621950312238152, −5.40883646224017742558833110929, −5.15737308702742439962830578182, −3.98545252174476450010379922674, −2.46980252567694702814165634370, −1.49406454726080304413932332146, 0,
1.49406454726080304413932332146, 2.46980252567694702814165634370, 3.98545252174476450010379922674, 5.15737308702742439962830578182, 5.40883646224017742558833110929, 6.69533610932342621950312238152, 7.53992226595484027540629580230, 8.269902934722587275039895689800, 9.361502774293671511491915199303
