| L(s) = 1 | − 3-s − 5-s + 7-s + 9-s − 2.82·11-s − 4.82·13-s + 15-s − 3.65·17-s − 2.82·19-s − 21-s + 4·23-s + 25-s − 27-s + 3.65·29-s − 2.82·31-s + 2.82·33-s − 35-s − 0.343·37-s + 4.82·39-s + 3.65·41-s + 9.65·43-s − 45-s + 11.3·47-s + 49-s + 3.65·51-s − 6.48·53-s + 2.82·55-s + ⋯ |
| L(s) = 1 | − 0.577·3-s − 0.447·5-s + 0.377·7-s + 0.333·9-s − 0.852·11-s − 1.33·13-s + 0.258·15-s − 0.886·17-s − 0.648·19-s − 0.218·21-s + 0.834·23-s + 0.200·25-s − 0.192·27-s + 0.679·29-s − 0.508·31-s + 0.492·33-s − 0.169·35-s − 0.0564·37-s + 0.773·39-s + 0.571·41-s + 1.47·43-s − 0.149·45-s + 1.65·47-s + 0.142·49-s + 0.512·51-s − 0.890·53-s + 0.381·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3360 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3360 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.9210950937\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.9210950937\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + T \) |
| 7 | \( 1 - T \) |
| good | 11 | \( 1 + 2.82T + 11T^{2} \) |
| 13 | \( 1 + 4.82T + 13T^{2} \) |
| 17 | \( 1 + 3.65T + 17T^{2} \) |
| 19 | \( 1 + 2.82T + 19T^{2} \) |
| 23 | \( 1 - 4T + 23T^{2} \) |
| 29 | \( 1 - 3.65T + 29T^{2} \) |
| 31 | \( 1 + 2.82T + 31T^{2} \) |
| 37 | \( 1 + 0.343T + 37T^{2} \) |
| 41 | \( 1 - 3.65T + 41T^{2} \) |
| 43 | \( 1 - 9.65T + 43T^{2} \) |
| 47 | \( 1 - 11.3T + 47T^{2} \) |
| 53 | \( 1 + 6.48T + 53T^{2} \) |
| 59 | \( 1 + 59T^{2} \) |
| 61 | \( 1 + 13.3T + 61T^{2} \) |
| 67 | \( 1 - 4T + 67T^{2} \) |
| 71 | \( 1 - 5.17T + 71T^{2} \) |
| 73 | \( 1 - 0.828T + 73T^{2} \) |
| 79 | \( 1 + 7.31T + 79T^{2} \) |
| 83 | \( 1 - 6.34T + 83T^{2} \) |
| 89 | \( 1 - 0.343T + 89T^{2} \) |
| 97 | \( 1 + 12.8T + 97T^{2} \) |
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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
−8.627423296709624906901707277084, −7.63546988022829457410782436562, −7.30421230268843493944950410527, −6.38769972009962741732693174729, −5.49161718975038720824069329861, −4.73463085838188686866199161074, −4.26315590316804096683565094468, −2.90266905089697740517576661562, −2.10296395468012365392438977092, −0.56468938323591597207244526122,
0.56468938323591597207244526122, 2.10296395468012365392438977092, 2.90266905089697740517576661562, 4.26315590316804096683565094468, 4.73463085838188686866199161074, 5.49161718975038720824069329861, 6.38769972009962741732693174729, 7.30421230268843493944950410527, 7.63546988022829457410782436562, 8.627423296709624906901707277084
