| L(s) = 1 | + 7-s − 11-s + 4·13-s − 6·17-s − 2·19-s + 23-s − 2·29-s − 31-s + 9·37-s − 6·41-s − 8·43-s − 8·47-s + 49-s + 10·53-s − 59-s − 2·61-s − 11·67-s − 11·71-s + 14·73-s − 77-s − 14·79-s + 4·83-s − 13·89-s + 4·91-s + 9·97-s + 101-s + 103-s + ⋯ |
| L(s) = 1 | + 0.377·7-s − 0.301·11-s + 1.10·13-s − 1.45·17-s − 0.458·19-s + 0.208·23-s − 0.371·29-s − 0.179·31-s + 1.47·37-s − 0.937·41-s − 1.21·43-s − 1.16·47-s + 1/7·49-s + 1.37·53-s − 0.130·59-s − 0.256·61-s − 1.34·67-s − 1.30·71-s + 1.63·73-s − 0.113·77-s − 1.57·79-s + 0.439·83-s − 1.37·89-s + 0.419·91-s + 0.913·97-s + 0.0995·101-s + 0.0985·103-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 69300 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 69300 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.708461312\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.708461312\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) |
| 7 | \( 1 - T \) |
| 11 | \( 1 + T \) |
| good | 13 | \( 1 - 4 T + p T^{2} \) |
| 17 | \( 1 + 6 T + p T^{2} \) |
| 19 | \( 1 + 2 T + p T^{2} \) |
| 23 | \( 1 - T + p T^{2} \) |
| 29 | \( 1 + 2 T + p T^{2} \) |
| 31 | \( 1 + T + p T^{2} \) |
| 37 | \( 1 - 9 T + p T^{2} \) |
| 41 | \( 1 + 6 T + p T^{2} \) |
| 43 | \( 1 + 8 T + p T^{2} \) |
| 47 | \( 1 + 8 T + p T^{2} \) |
| 53 | \( 1 - 10 T + p T^{2} \) |
| 59 | \( 1 + T + p T^{2} \) |
| 61 | \( 1 + 2 T + p T^{2} \) |
| 67 | \( 1 + 11 T + p T^{2} \) |
| 71 | \( 1 + 11 T + p T^{2} \) |
| 73 | \( 1 - 14 T + p T^{2} \) |
| 79 | \( 1 + 14 T + p T^{2} \) |
| 83 | \( 1 - 4 T + p T^{2} \) |
| 89 | \( 1 + 13 T + p T^{2} \) |
| 97 | \( 1 - 9 T + p T^{2} \) |
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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
−14.10006397317070, −13.45399288223861, −13.20907048765918, −12.91581454887176, −12.01934480042085, −11.56732645467151, −11.12105454618601, −10.75289804066272, −10.15295100933527, −9.601203882028886, −8.840839400049176, −8.620520585039109, −8.109261336439878, −7.432996853483601, −6.857619009413052, −6.317281268877471, −5.867840107680056, −5.137487182294903, −4.556342834510006, −4.096919880894109, −3.385379271760568, −2.723745978102176, −1.961136488840066, −1.441152775358302, −0.4298858375349706,
0.4298858375349706, 1.441152775358302, 1.961136488840066, 2.723745978102176, 3.385379271760568, 4.096919880894109, 4.556342834510006, 5.137487182294903, 5.867840107680056, 6.317281268877471, 6.857619009413052, 7.432996853483601, 8.109261336439878, 8.620520585039109, 8.840839400049176, 9.601203882028886, 10.15295100933527, 10.75289804066272, 11.12105454618601, 11.56732645467151, 12.01934480042085, 12.91581454887176, 13.20907048765918, 13.45399288223861, 14.10006397317070
