| L(s) = 1 | + 2.03·3-s + 1.15·9-s + 1.04·11-s − 1.15·13-s − 6.81·17-s + 4.75·19-s − 3.19·23-s − 3.76·27-s + 5.14·29-s + 5.57·31-s + 2.13·33-s − 4.91·37-s − 2.35·39-s − 9.68·41-s − 7.44·43-s − 9.29·47-s − 13.9·51-s + 5.60·53-s + 9.68·57-s − 10.8·59-s + 3.33·61-s − 5.45·67-s − 6.50·69-s − 4.10·71-s − 6.46·73-s + 0.612·79-s − 11.1·81-s + ⋯ |
| L(s) = 1 | + 1.17·3-s + 0.384·9-s + 0.316·11-s − 0.320·13-s − 1.65·17-s + 1.08·19-s − 0.665·23-s − 0.723·27-s + 0.954·29-s + 1.00·31-s + 0.371·33-s − 0.808·37-s − 0.376·39-s − 1.51·41-s − 1.13·43-s − 1.35·47-s − 1.94·51-s + 0.769·53-s + 1.28·57-s − 1.41·59-s + 0.426·61-s − 0.665·67-s − 0.783·69-s − 0.487·71-s − 0.757·73-s + 0.0689·79-s − 1.23·81-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 9800 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 9800 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 5 | \( 1 \) |
| 7 | \( 1 \) |
| good | 3 | \( 1 - 2.03T + 3T^{2} \) |
| 11 | \( 1 - 1.04T + 11T^{2} \) |
| 13 | \( 1 + 1.15T + 13T^{2} \) |
| 17 | \( 1 + 6.81T + 17T^{2} \) |
| 19 | \( 1 - 4.75T + 19T^{2} \) |
| 23 | \( 1 + 3.19T + 23T^{2} \) |
| 29 | \( 1 - 5.14T + 29T^{2} \) |
| 31 | \( 1 - 5.57T + 31T^{2} \) |
| 37 | \( 1 + 4.91T + 37T^{2} \) |
| 41 | \( 1 + 9.68T + 41T^{2} \) |
| 43 | \( 1 + 7.44T + 43T^{2} \) |
| 47 | \( 1 + 9.29T + 47T^{2} \) |
| 53 | \( 1 - 5.60T + 53T^{2} \) |
| 59 | \( 1 + 10.8T + 59T^{2} \) |
| 61 | \( 1 - 3.33T + 61T^{2} \) |
| 67 | \( 1 + 5.45T + 67T^{2} \) |
| 71 | \( 1 + 4.10T + 71T^{2} \) |
| 73 | \( 1 + 6.46T + 73T^{2} \) |
| 79 | \( 1 - 0.612T + 79T^{2} \) |
| 83 | \( 1 + 0.275T + 83T^{2} \) |
| 89 | \( 1 - 17.2T + 89T^{2} \) |
| 97 | \( 1 - 7.59T + 97T^{2} \) |
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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
−7.42128810321885828825363453845, −6.72460911063100748158838770944, −6.16356369191253804472536472105, −5.05627136798787394406433855140, −4.54014561127937240035514709228, −3.59890517414622758599108564672, −3.04207111986880176376490854988, −2.25784621159464574823406103474, −1.50111039642258913485795151579, 0,
1.50111039642258913485795151579, 2.25784621159464574823406103474, 3.04207111986880176376490854988, 3.59890517414622758599108564672, 4.54014561127937240035514709228, 5.05627136798787394406433855140, 6.16356369191253804472536472105, 6.72460911063100748158838770944, 7.42128810321885828825363453845
