| L(s) = 1 | + 5-s + 3.44·7-s − 2.44·11-s − 4.44·13-s − 0.550·17-s + 2.44·19-s − 23-s + 25-s − 7.89·29-s − 7.89·31-s + 3.44·35-s + 8.34·37-s − 1.89·41-s − 0.898·43-s − 1.55·47-s + 4.89·49-s + 7.44·53-s − 2.44·55-s − 59-s + 4.44·61-s − 4.44·65-s − 9.44·67-s − 5.89·71-s − 3.55·73-s − 8.44·77-s − 15.4·83-s − 0.550·85-s + ⋯ |
| L(s) = 1 | + 0.447·5-s + 1.30·7-s − 0.738·11-s − 1.23·13-s − 0.133·17-s + 0.561·19-s − 0.208·23-s + 0.200·25-s − 1.46·29-s − 1.41·31-s + 0.583·35-s + 1.37·37-s − 0.296·41-s − 0.137·43-s − 0.226·47-s + 0.699·49-s + 1.02·53-s − 0.330·55-s − 0.130·59-s + 0.569·61-s − 0.551·65-s − 1.15·67-s − 0.700·71-s − 0.415·73-s − 0.962·77-s − 1.69·83-s − 0.0597·85-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8280 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8280 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 5 | \( 1 - T \) |
| 23 | \( 1 + T \) |
| good | 7 | \( 1 - 3.44T + 7T^{2} \) |
| 11 | \( 1 + 2.44T + 11T^{2} \) |
| 13 | \( 1 + 4.44T + 13T^{2} \) |
| 17 | \( 1 + 0.550T + 17T^{2} \) |
| 19 | \( 1 - 2.44T + 19T^{2} \) |
| 29 | \( 1 + 7.89T + 29T^{2} \) |
| 31 | \( 1 + 7.89T + 31T^{2} \) |
| 37 | \( 1 - 8.34T + 37T^{2} \) |
| 41 | \( 1 + 1.89T + 41T^{2} \) |
| 43 | \( 1 + 0.898T + 43T^{2} \) |
| 47 | \( 1 + 1.55T + 47T^{2} \) |
| 53 | \( 1 - 7.44T + 53T^{2} \) |
| 59 | \( 1 + T + 59T^{2} \) |
| 61 | \( 1 - 4.44T + 61T^{2} \) |
| 67 | \( 1 + 9.44T + 67T^{2} \) |
| 71 | \( 1 + 5.89T + 71T^{2} \) |
| 73 | \( 1 + 3.55T + 73T^{2} \) |
| 79 | \( 1 + 79T^{2} \) |
| 83 | \( 1 + 15.4T + 83T^{2} \) |
| 89 | \( 1 + 8.89T + 89T^{2} \) |
| 97 | \( 1 + 1.10T + 97T^{2} \) |
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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
−7.45533438511854099882657417762, −7.05074059766755066945718435990, −5.76237648170867533742697236550, −5.44018893749979220373850561463, −4.75064185031391896362260835966, −4.04792494046729827266251847162, −2.89413608040190366073225310833, −2.16394552726948847362326492208, −1.44476742938315639647106323083, 0,
1.44476742938315639647106323083, 2.16394552726948847362326492208, 2.89413608040190366073225310833, 4.04792494046729827266251847162, 4.75064185031391896362260835966, 5.44018893749979220373850561463, 5.76237648170867533742697236550, 7.05074059766755066945718435990, 7.45533438511854099882657417762
