| L(s) = 1 | − 3-s + 5-s + 9-s + 13-s − 15-s + 2·17-s − 4·19-s − 4·23-s + 25-s − 27-s + 6·29-s − 8·31-s + 6·37-s − 39-s − 2·41-s + 4·43-s + 45-s − 7·49-s − 2·51-s − 6·53-s + 4·57-s + 2·61-s + 65-s + 8·67-s + 4·69-s + 6·73-s − 75-s + ⋯ |
| L(s) = 1 | − 0.577·3-s + 0.447·5-s + 1/3·9-s + 0.277·13-s − 0.258·15-s + 0.485·17-s − 0.917·19-s − 0.834·23-s + 1/5·25-s − 0.192·27-s + 1.11·29-s − 1.43·31-s + 0.986·37-s − 0.160·39-s − 0.312·41-s + 0.609·43-s + 0.149·45-s − 49-s − 0.280·51-s − 0.824·53-s + 0.529·57-s + 0.256·61-s + 0.124·65-s + 0.977·67-s + 0.481·69-s + 0.702·73-s − 0.115·75-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 12480 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 12480 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 5 | \( 1 - T \) |
| 13 | \( 1 - T \) |
| good | 7 | \( 1 + p T^{2} \) |
| 11 | \( 1 + p T^{2} \) |
| 17 | \( 1 - 2 T + p T^{2} \) |
| 19 | \( 1 + 4 T + p T^{2} \) |
| 23 | \( 1 + 4 T + p T^{2} \) |
| 29 | \( 1 - 6 T + p T^{2} \) |
| 31 | \( 1 + 8 T + p T^{2} \) |
| 37 | \( 1 - 6 T + p T^{2} \) |
| 41 | \( 1 + 2 T + p T^{2} \) |
| 43 | \( 1 - 4 T + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 + 6 T + p T^{2} \) |
| 59 | \( 1 + p T^{2} \) |
| 61 | \( 1 - 2 T + p T^{2} \) |
| 67 | \( 1 - 8 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 - 6 T + p T^{2} \) |
| 79 | \( 1 - 4 T + p T^{2} \) |
| 83 | \( 1 + 12 T + p T^{2} \) |
| 89 | \( 1 - 6 T + p T^{2} \) |
| 97 | \( 1 + 10 T + p T^{2} \) |
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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
−16.45607218535437, −16.24582178946437, −15.57014258715666, −14.83430177461578, −14.36815304473073, −13.80235199256147, −13.06160610318106, −12.65583635912552, −12.09399975668941, −11.41339661881216, −10.78819223189630, −10.41882893372739, −9.590998211350889, −9.281822515432296, −8.204453052637069, −7.985101238854525, −6.884805991537749, −6.514674413518212, −5.775043856683045, −5.300912424763365, −4.431001097290510, −3.846376444387486, −2.870329639648229, −2.002229364056418, −1.176603716546758, 0,
1.176603716546758, 2.002229364056418, 2.870329639648229, 3.846376444387486, 4.431001097290510, 5.300912424763365, 5.775043856683045, 6.514674413518212, 6.884805991537749, 7.985101238854525, 8.204453052637069, 9.281822515432296, 9.590998211350889, 10.41882893372739, 10.78819223189630, 11.41339661881216, 12.09399975668941, 12.65583635912552, 13.06160610318106, 13.80235199256147, 14.36815304473073, 14.83430177461578, 15.57014258715666, 16.24582178946437, 16.45607218535437
