| L(s) = 1 | + 4·7-s − 4·13-s − 6·17-s − 19-s − 6·23-s − 5·25-s − 6·29-s − 2·31-s − 4·37-s − 6·41-s + 4·43-s + 6·47-s + 9·49-s − 6·53-s − 12·59-s + 14·61-s − 8·67-s + 14·73-s + 10·79-s − 12·83-s + 6·89-s − 16·91-s − 10·97-s + 10·103-s + 12·107-s − 16·109-s + 18·113-s + ⋯ |
| L(s) = 1 | + 1.51·7-s − 1.10·13-s − 1.45·17-s − 0.229·19-s − 1.25·23-s − 25-s − 1.11·29-s − 0.359·31-s − 0.657·37-s − 0.937·41-s + 0.609·43-s + 0.875·47-s + 9/7·49-s − 0.824·53-s − 1.56·59-s + 1.79·61-s − 0.977·67-s + 1.63·73-s + 1.12·79-s − 1.31·83-s + 0.635·89-s − 1.67·91-s − 1.01·97-s + 0.985·103-s + 1.16·107-s − 1.53·109-s + 1.69·113-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2736 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2736 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 19 | \( 1 + T \) |
| good | 5 | \( 1 + p T^{2} \) |
| 7 | \( 1 - 4 T + p T^{2} \) |
| 11 | \( 1 + p T^{2} \) |
| 13 | \( 1 + 4 T + p T^{2} \) |
| 17 | \( 1 + 6 T + p T^{2} \) |
| 23 | \( 1 + 6 T + p T^{2} \) |
| 29 | \( 1 + 6 T + p T^{2} \) |
| 31 | \( 1 + 2 T + p T^{2} \) |
| 37 | \( 1 + 4 T + p T^{2} \) |
| 41 | \( 1 + 6 T + p T^{2} \) |
| 43 | \( 1 - 4 T + p T^{2} \) |
| 47 | \( 1 - 6 T + p T^{2} \) |
| 53 | \( 1 + 6 T + p T^{2} \) |
| 59 | \( 1 + 12 T + p T^{2} \) |
| 61 | \( 1 - 14 T + p T^{2} \) |
| 67 | \( 1 + 8 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 - 14 T + p T^{2} \) |
| 79 | \( 1 - 10 T + p T^{2} \) |
| 83 | \( 1 + 12 T + p T^{2} \) |
| 89 | \( 1 - 6 T + p T^{2} \) |
| 97 | \( 1 + 10 T + p T^{2} \) |
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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
−8.359688629223917749842479932615, −7.72795819597559924442133345234, −7.10678703287277123493279316153, −6.07507983377375758209992822317, −5.21403289865941834256074312934, −4.54790713149621293279441185278, −3.81156610301665935267786910207, −2.27147129954073815802695201168, −1.81801100280978906223857531826, 0,
1.81801100280978906223857531826, 2.27147129954073815802695201168, 3.81156610301665935267786910207, 4.54790713149621293279441185278, 5.21403289865941834256074312934, 6.07507983377375758209992822317, 7.10678703287277123493279316153, 7.72795819597559924442133345234, 8.359688629223917749842479932615
