| L(s) = 1 | − 3-s + 7-s + 9-s + 1.05·11-s + 3.55·13-s − 4.49·17-s + 19-s − 21-s + 7.43·23-s − 5·25-s − 27-s − 9.55·29-s − 6.61·31-s − 1.05·33-s − 8.61·37-s − 3.55·39-s − 0.117·41-s + 1.88·43-s + 0.941·47-s + 49-s + 4.49·51-s − 2.44·53-s − 57-s − 12.9·59-s + 6.99·61-s + 63-s − 0.824·67-s + ⋯ |
| L(s) = 1 | − 0.577·3-s + 0.377·7-s + 0.333·9-s + 0.319·11-s + 0.986·13-s − 1.09·17-s + 0.229·19-s − 0.218·21-s + 1.55·23-s − 25-s − 0.192·27-s − 1.77·29-s − 1.18·31-s − 0.184·33-s − 1.41·37-s − 0.569·39-s − 0.0183·41-s + 0.287·43-s + 0.137·47-s + 0.142·49-s + 0.629·51-s − 0.335·53-s − 0.132·57-s − 1.69·59-s + 0.895·61-s + 0.125·63-s − 0.100·67-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6384 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6384 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 7 | \( 1 - T \) |
| 19 | \( 1 - T \) |
| good | 5 | \( 1 + 5T^{2} \) |
| 11 | \( 1 - 1.05T + 11T^{2} \) |
| 13 | \( 1 - 3.55T + 13T^{2} \) |
| 17 | \( 1 + 4.49T + 17T^{2} \) |
| 23 | \( 1 - 7.43T + 23T^{2} \) |
| 29 | \( 1 + 9.55T + 29T^{2} \) |
| 31 | \( 1 + 6.61T + 31T^{2} \) |
| 37 | \( 1 + 8.61T + 37T^{2} \) |
| 41 | \( 1 + 0.117T + 41T^{2} \) |
| 43 | \( 1 - 1.88T + 43T^{2} \) |
| 47 | \( 1 - 0.941T + 47T^{2} \) |
| 53 | \( 1 + 2.44T + 53T^{2} \) |
| 59 | \( 1 + 12.9T + 59T^{2} \) |
| 61 | \( 1 - 6.99T + 61T^{2} \) |
| 67 | \( 1 + 0.824T + 67T^{2} \) |
| 71 | \( 1 + 1.43T + 71T^{2} \) |
| 73 | \( 1 - 0.117T + 73T^{2} \) |
| 79 | \( 1 - 3.67T + 79T^{2} \) |
| 83 | \( 1 - 9.67T + 83T^{2} \) |
| 89 | \( 1 - 5.11T + 89T^{2} \) |
| 97 | \( 1 + 12.0T + 97T^{2} \) |
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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
−7.50598385616260073530011716649, −6.99646017230105862538680907687, −6.21899490251417499817202657669, −5.53974340265759809372476664087, −4.90996154449316727413411487290, −3.98996382273669536564610202251, −3.41131698080331869757911325606, −2.07848062430910655674147956471, −1.33652001813755092086703961338, 0,
1.33652001813755092086703961338, 2.07848062430910655674147956471, 3.41131698080331869757911325606, 3.98996382273669536564610202251, 4.90996154449316727413411487290, 5.53974340265759809372476664087, 6.21899490251417499817202657669, 6.99646017230105862538680907687, 7.50598385616260073530011716649
