| L(s) = 1 | − 2·5-s − 3·9-s − 2·13-s − 17-s + 4·19-s − 4·23-s − 25-s − 6·29-s + 4·31-s + 2·37-s + 6·41-s + 4·43-s + 6·45-s − 6·53-s + 12·59-s − 10·61-s + 4·65-s + 4·67-s + 4·71-s + 6·73-s − 12·79-s + 9·81-s + 4·83-s + 2·85-s − 10·89-s − 8·95-s − 2·97-s + ⋯ |
| L(s) = 1 | − 0.894·5-s − 9-s − 0.554·13-s − 0.242·17-s + 0.917·19-s − 0.834·23-s − 1/5·25-s − 1.11·29-s + 0.718·31-s + 0.328·37-s + 0.937·41-s + 0.609·43-s + 0.894·45-s − 0.824·53-s + 1.56·59-s − 1.28·61-s + 0.496·65-s + 0.488·67-s + 0.474·71-s + 0.702·73-s − 1.35·79-s + 81-s + 0.439·83-s + 0.216·85-s − 1.05·89-s − 0.820·95-s − 0.203·97-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 53312 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 53312 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 7 | \( 1 \) |
| 17 | \( 1 + T \) |
| good | 3 | \( 1 + p T^{2} \) |
| 5 | \( 1 + 2 T + p T^{2} \) |
| 11 | \( 1 + p T^{2} \) |
| 13 | \( 1 + 2 T + p T^{2} \) |
| 19 | \( 1 - 4 T + p T^{2} \) |
| 23 | \( 1 + 4 T + p T^{2} \) |
| 29 | \( 1 + 6 T + p T^{2} \) |
| 31 | \( 1 - 4 T + p T^{2} \) |
| 37 | \( 1 - 2 T + p T^{2} \) |
| 41 | \( 1 - 6 T + p T^{2} \) |
| 43 | \( 1 - 4 T + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 + 6 T + p T^{2} \) |
| 59 | \( 1 - 12 T + p T^{2} \) |
| 61 | \( 1 + 10 T + p T^{2} \) |
| 67 | \( 1 - 4 T + p T^{2} \) |
| 71 | \( 1 - 4 T + p T^{2} \) |
| 73 | \( 1 - 6 T + p T^{2} \) |
| 79 | \( 1 + 12 T + p T^{2} \) |
| 83 | \( 1 - 4 T + p T^{2} \) |
| 89 | \( 1 + 10 T + p T^{2} \) |
| 97 | \( 1 + 2 T + p T^{2} \) |
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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
−14.63432322671384, −14.24107735002637, −13.81372283977075, −13.16146737281556, −12.55646892518444, −12.07837077966596, −11.52813754054509, −11.35873398509629, −10.70493113670878, −10.03944742334240, −9.414365874274381, −9.102764766978692, −8.234132863811276, −7.938144537084353, −7.500866096662901, −6.836234408639757, −6.150697350661031, −5.575025305097962, −5.113754372098505, −4.228395237192728, −3.915948325140020, −3.086390434708327, −2.607603860413929, −1.805791831212936, −0.7290279788925337, 0,
0.7290279788925337, 1.805791831212936, 2.607603860413929, 3.086390434708327, 3.915948325140020, 4.228395237192728, 5.113754372098505, 5.575025305097962, 6.150697350661031, 6.836234408639757, 7.500866096662901, 7.938144537084353, 8.234132863811276, 9.102764766978692, 9.414365874274381, 10.03944742334240, 10.70493113670878, 11.35873398509629, 11.52813754054509, 12.07837077966596, 12.55646892518444, 13.16146737281556, 13.81372283977075, 14.24107735002637, 14.63432322671384
