| L(s) = 1 | − 2·3-s − 2·4-s − 5-s + 3·7-s + 9-s + 11-s + 4·12-s + 2·15-s + 4·16-s + 17-s + 2·20-s − 6·21-s − 2·23-s + 25-s + 4·27-s − 6·28-s + 3·29-s − 10·31-s − 2·33-s − 3·35-s − 2·36-s − 4·37-s − 41-s − 8·43-s − 2·44-s − 45-s − 3·47-s + ⋯ |
| L(s) = 1 | − 1.15·3-s − 4-s − 0.447·5-s + 1.13·7-s + 1/3·9-s + 0.301·11-s + 1.15·12-s + 0.516·15-s + 16-s + 0.242·17-s + 0.447·20-s − 1.30·21-s − 0.417·23-s + 1/5·25-s + 0.769·27-s − 1.13·28-s + 0.557·29-s − 1.79·31-s − 0.348·33-s − 0.507·35-s − 1/3·36-s − 0.657·37-s − 0.156·41-s − 1.21·43-s − 0.301·44-s − 0.149·45-s − 0.437·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 935 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 935 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 5 | \( 1 + T \) |
| 11 | \( 1 - T \) |
| 17 | \( 1 - T \) |
| good | 2 | \( 1 + p T^{2} \) |
| 3 | \( 1 + 2 T + p T^{2} \) |
| 7 | \( 1 - 3 T + p T^{2} \) |
| 13 | \( 1 + p T^{2} \) |
| 19 | \( 1 + p T^{2} \) |
| 23 | \( 1 + 2 T + p T^{2} \) |
| 29 | \( 1 - 3 T + p T^{2} \) |
| 31 | \( 1 + 10 T + p T^{2} \) |
| 37 | \( 1 + 4 T + p T^{2} \) |
| 41 | \( 1 + T + p T^{2} \) |
| 43 | \( 1 + 8 T + p T^{2} \) |
| 47 | \( 1 + 3 T + p T^{2} \) |
| 53 | \( 1 + 9 T + p T^{2} \) |
| 59 | \( 1 - T + p T^{2} \) |
| 61 | \( 1 - 2 T + p T^{2} \) |
| 67 | \( 1 + 3 T + p T^{2} \) |
| 71 | \( 1 - 2 T + p T^{2} \) |
| 73 | \( 1 + 11 T + p T^{2} \) |
| 79 | \( 1 + p T^{2} \) |
| 83 | \( 1 - 6 T + p T^{2} \) |
| 89 | \( 1 + 7 T + p T^{2} \) |
| 97 | \( 1 + 12 T + p T^{2} \) |
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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
−9.724435653100562883024060078942, −8.681915110330252085816967149299, −8.096518380999893458145430301922, −7.09213040713891537451754400928, −5.93320083884330048839746032322, −5.13333593825616424217456727852, −4.57442960156456663256964569172, −3.49946827040299524410574434522, −1.45862075631267716833396345264, 0,
1.45862075631267716833396345264, 3.49946827040299524410574434522, 4.57442960156456663256964569172, 5.13333593825616424217456727852, 5.93320083884330048839746032322, 7.09213040713891537451754400928, 8.096518380999893458145430301922, 8.681915110330252085816967149299, 9.724435653100562883024060078942
