| L(s) = 1 | − 1.41·3-s − 3.41·5-s + 7-s − 0.999·9-s − 2·11-s + 2.82·13-s + 4.82·15-s + 5.41·17-s − 2·19-s − 1.41·21-s + 23-s + 6.65·25-s + 5.65·27-s + 8.48·29-s − 0.585·31-s + 2.82·33-s − 3.41·35-s − 6.48·37-s − 4.00·39-s + 6·41-s − 8·43-s + 3.41·45-s − 7.41·47-s + 49-s − 7.65·51-s + 4.82·53-s + 6.82·55-s + ⋯ |
| L(s) = 1 | − 0.816·3-s − 1.52·5-s + 0.377·7-s − 0.333·9-s − 0.603·11-s + 0.784·13-s + 1.24·15-s + 1.31·17-s − 0.458·19-s − 0.308·21-s + 0.208·23-s + 1.33·25-s + 1.08·27-s + 1.57·29-s − 0.105·31-s + 0.492·33-s − 0.577·35-s − 1.06·37-s − 0.640·39-s + 0.937·41-s − 1.21·43-s + 0.508·45-s − 1.08·47-s + 0.142·49-s − 1.07·51-s + 0.663·53-s + 0.920·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2576 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2576 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 7 | \( 1 - T \) |
| 23 | \( 1 - T \) |
| good | 3 | \( 1 + 1.41T + 3T^{2} \) |
| 5 | \( 1 + 3.41T + 5T^{2} \) |
| 11 | \( 1 + 2T + 11T^{2} \) |
| 13 | \( 1 - 2.82T + 13T^{2} \) |
| 17 | \( 1 - 5.41T + 17T^{2} \) |
| 19 | \( 1 + 2T + 19T^{2} \) |
| 29 | \( 1 - 8.48T + 29T^{2} \) |
| 31 | \( 1 + 0.585T + 31T^{2} \) |
| 37 | \( 1 + 6.48T + 37T^{2} \) |
| 41 | \( 1 - 6T + 41T^{2} \) |
| 43 | \( 1 + 8T + 43T^{2} \) |
| 47 | \( 1 + 7.41T + 47T^{2} \) |
| 53 | \( 1 - 4.82T + 53T^{2} \) |
| 59 | \( 1 + 0.928T + 59T^{2} \) |
| 61 | \( 1 + 10.2T + 61T^{2} \) |
| 67 | \( 1 + 11.6T + 67T^{2} \) |
| 71 | \( 1 + 4.48T + 71T^{2} \) |
| 73 | \( 1 - 10.4T + 73T^{2} \) |
| 79 | \( 1 - 6.48T + 79T^{2} \) |
| 83 | \( 1 - 0.828T + 83T^{2} \) |
| 89 | \( 1 - 15.0T + 89T^{2} \) |
| 97 | \( 1 + 15.5T + 97T^{2} \) |
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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
−8.145963778671619496752410573347, −8.024543867098049819735400746888, −6.97038759644118903543922917754, −6.18766800139152395534706695268, −5.26649405584074398924334957605, −4.65761069431831092208244582976, −3.66149963836100476446273527347, −2.90919312864122177717822616779, −1.15980550787076194876277007286, 0,
1.15980550787076194876277007286, 2.90919312864122177717822616779, 3.66149963836100476446273527347, 4.65761069431831092208244582976, 5.26649405584074398924334957605, 6.18766800139152395534706695268, 6.97038759644118903543922917754, 8.024543867098049819735400746888, 8.145963778671619496752410573347
