| L(s) = 1 | + 2-s + 4-s + 7-s + 8-s − 1.16·11-s + 2.16·13-s + 14-s + 16-s − 5·17-s − 0.837·19-s − 1.16·22-s + 0.162·23-s + 2.16·26-s + 28-s − 2.16·29-s − 8.48·31-s + 32-s − 5·34-s − 9.16·37-s − 0.837·38-s + 2.32·41-s − 7·43-s − 1.16·44-s + 0.162·46-s + 2·47-s + 49-s + 2.16·52-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 0.5·4-s + 0.377·7-s + 0.353·8-s − 0.350·11-s + 0.599·13-s + 0.267·14-s + 0.250·16-s − 1.21·17-s − 0.192·19-s − 0.247·22-s + 0.0338·23-s + 0.424·26-s + 0.188·28-s − 0.401·29-s − 1.52·31-s + 0.176·32-s − 0.857·34-s − 1.50·37-s − 0.135·38-s + 0.363·41-s − 1.06·43-s − 0.175·44-s + 0.0239·46-s + 0.291·47-s + 0.142·49-s + 0.299·52-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 9450 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 9450 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) |
| 3 | \( 1 \) |
| 5 | \( 1 \) |
| 7 | \( 1 - T \) |
| good | 11 | \( 1 + 1.16T + 11T^{2} \) |
| 13 | \( 1 - 2.16T + 13T^{2} \) |
| 17 | \( 1 + 5T + 17T^{2} \) |
| 19 | \( 1 + 0.837T + 19T^{2} \) |
| 23 | \( 1 - 0.162T + 23T^{2} \) |
| 29 | \( 1 + 2.16T + 29T^{2} \) |
| 31 | \( 1 + 8.48T + 31T^{2} \) |
| 37 | \( 1 + 9.16T + 37T^{2} \) |
| 41 | \( 1 - 2.32T + 41T^{2} \) |
| 43 | \( 1 + 7T + 43T^{2} \) |
| 47 | \( 1 - 2T + 47T^{2} \) |
| 53 | \( 1 - 12.4T + 53T^{2} \) |
| 59 | \( 1 + 7T + 59T^{2} \) |
| 61 | \( 1 + 8.32T + 61T^{2} \) |
| 67 | \( 1 + 15.3T + 67T^{2} \) |
| 71 | \( 1 - 6.48T + 71T^{2} \) |
| 73 | \( 1 + 10T + 73T^{2} \) |
| 79 | \( 1 - 6.83T + 79T^{2} \) |
| 83 | \( 1 - 8.32T + 83T^{2} \) |
| 89 | \( 1 + 15.3T + 89T^{2} \) |
| 97 | \( 1 + 1.16T + 97T^{2} \) |
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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
−7.24123915431750675823216960011, −6.65413419083383114114096329771, −5.86737116452966091414934787619, −5.29635600342569658601727429274, −4.55867102477661273753914422532, −3.89595472700994267998189889485, −3.16029558051035523207604603660, −2.19045379755382407101031136167, −1.52907631095352486749767915256, 0,
1.52907631095352486749767915256, 2.19045379755382407101031136167, 3.16029558051035523207604603660, 3.89595472700994267998189889485, 4.55867102477661273753914422532, 5.29635600342569658601727429274, 5.86737116452966091414934787619, 6.65413419083383114114096329771, 7.24123915431750675823216960011
