| L(s) = 1 | − 1.41·3-s + 5-s − 2.41·7-s − 0.999·9-s + 0.828·11-s + 3.41·13-s − 1.41·15-s + 2.24·17-s − 1.58·19-s + 3.41·21-s − 2.58·23-s − 4·25-s + 5.65·27-s + 7.07·29-s − 31-s − 1.17·33-s − 2.41·35-s + 0.343·37-s − 4.82·39-s − 6.65·41-s − 7.89·43-s − 0.999·45-s − 1.17·47-s − 1.17·49-s − 3.17·51-s + 4·53-s + 0.828·55-s + ⋯ |
| L(s) = 1 | − 0.816·3-s + 0.447·5-s − 0.912·7-s − 0.333·9-s + 0.249·11-s + 0.946·13-s − 0.365·15-s + 0.543·17-s − 0.363·19-s + 0.745·21-s − 0.539·23-s − 0.800·25-s + 1.08·27-s + 1.31·29-s − 0.179·31-s − 0.203·33-s − 0.408·35-s + 0.0564·37-s − 0.773·39-s − 1.03·41-s − 1.20·43-s − 0.149·45-s − 0.170·47-s − 0.167·49-s − 0.444·51-s + 0.549·53-s + 0.111·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1984 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1984 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 31 | \( 1 + T \) |
| good | 3 | \( 1 + 1.41T + 3T^{2} \) |
| 5 | \( 1 - T + 5T^{2} \) |
| 7 | \( 1 + 2.41T + 7T^{2} \) |
| 11 | \( 1 - 0.828T + 11T^{2} \) |
| 13 | \( 1 - 3.41T + 13T^{2} \) |
| 17 | \( 1 - 2.24T + 17T^{2} \) |
| 19 | \( 1 + 1.58T + 19T^{2} \) |
| 23 | \( 1 + 2.58T + 23T^{2} \) |
| 29 | \( 1 - 7.07T + 29T^{2} \) |
| 37 | \( 1 - 0.343T + 37T^{2} \) |
| 41 | \( 1 + 6.65T + 41T^{2} \) |
| 43 | \( 1 + 7.89T + 43T^{2} \) |
| 47 | \( 1 + 1.17T + 47T^{2} \) |
| 53 | \( 1 - 4T + 53T^{2} \) |
| 59 | \( 1 + 10.0T + 59T^{2} \) |
| 61 | \( 1 - 1.75T + 61T^{2} \) |
| 67 | \( 1 + 6T + 67T^{2} \) |
| 71 | \( 1 + 3.58T + 71T^{2} \) |
| 73 | \( 1 + 11.6T + 73T^{2} \) |
| 79 | \( 1 - 7.41T + 79T^{2} \) |
| 83 | \( 1 + 4.82T + 83T^{2} \) |
| 89 | \( 1 - 3.75T + 89T^{2} \) |
| 97 | \( 1 + 9.48T + 97T^{2} \) |
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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
−8.772516821437678610627736257945, −8.125423230465924281134013705342, −6.88881985717632081362274528505, −6.22132513728934019777622130942, −5.83979071243257222040069844014, −4.86155876130595018282449515901, −3.73648815920486104577104891507, −2.86848557514914469203388406011, −1.44947404486702301485169284316, 0,
1.44947404486702301485169284316, 2.86848557514914469203388406011, 3.73648815920486104577104891507, 4.86155876130595018282449515901, 5.83979071243257222040069844014, 6.22132513728934019777622130942, 6.88881985717632081362274528505, 8.125423230465924281134013705342, 8.772516821437678610627736257945
