| L(s) = 1 | + 2-s − 3-s + 4-s + 5-s − 6-s − 7-s + 8-s + 9-s + 10-s + 2·11-s − 12-s + 7·13-s − 14-s − 15-s + 16-s + 18-s + 6·19-s + 20-s + 21-s + 2·22-s − 3·23-s − 24-s − 4·25-s + 7·26-s − 27-s − 28-s − 2·29-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 0.577·3-s + 1/2·4-s + 0.447·5-s − 0.408·6-s − 0.377·7-s + 0.353·8-s + 1/3·9-s + 0.316·10-s + 0.603·11-s − 0.288·12-s + 1.94·13-s − 0.267·14-s − 0.258·15-s + 1/4·16-s + 0.235·18-s + 1.37·19-s + 0.223·20-s + 0.218·21-s + 0.426·22-s − 0.625·23-s − 0.204·24-s − 4/5·25-s + 1.37·26-s − 0.192·27-s − 0.188·28-s − 0.371·29-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 195942 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 195942 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(4.743667035\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.743667035\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - T \) |
| 3 | \( 1 + T \) |
| 17 | \( 1 \) |
| 113 | \( 1 + T \) |
| good | 5 | \( 1 - T + p T^{2} \) |
| 7 | \( 1 + T + p T^{2} \) |
| 11 | \( 1 - 2 T + p T^{2} \) |
| 13 | \( 1 - 7 T + p T^{2} \) |
| 19 | \( 1 - 6 T + p T^{2} \) |
| 23 | \( 1 + 3 T + p T^{2} \) |
| 29 | \( 1 + 2 T + p T^{2} \) |
| 31 | \( 1 - 3 T + p T^{2} \) |
| 37 | \( 1 - 4 T + p T^{2} \) |
| 41 | \( 1 + p T^{2} \) |
| 43 | \( 1 - 2 T + p T^{2} \) |
| 47 | \( 1 + 8 T + p T^{2} \) |
| 53 | \( 1 + 2 T + p T^{2} \) |
| 59 | \( 1 + 3 T + p T^{2} \) |
| 61 | \( 1 - T + p T^{2} \) |
| 67 | \( 1 + 2 T + p T^{2} \) |
| 71 | \( 1 - 5 T + p T^{2} \) |
| 73 | \( 1 + 4 T + p T^{2} \) |
| 79 | \( 1 - 4 T + p T^{2} \) |
| 83 | \( 1 + p T^{2} \) |
| 89 | \( 1 - 6 T + p T^{2} \) |
| 97 | \( 1 - 14 T + p T^{2} \) |
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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
−13.14228351408150, −12.73127484924863, −12.07348564695303, −11.63639696953220, −11.36546663542228, −10.92579679147785, −10.27997327443798, −9.867637514837967, −9.381025951525982, −8.948735159406311, −8.149076523982457, −7.834205709446429, −7.171810601636657, −6.457850704975598, −6.269555214862620, −5.883855003843390, −5.370932273476334, −4.777731800692411, −4.151019138024256, −3.594289465288386, −3.339006541410844, −2.528059028638369, −1.657801396360715, −1.347889484884899, −0.5825158965799984,
0.5825158965799984, 1.347889484884899, 1.657801396360715, 2.528059028638369, 3.339006541410844, 3.594289465288386, 4.151019138024256, 4.777731800692411, 5.370932273476334, 5.883855003843390, 6.269555214862620, 6.457850704975598, 7.171810601636657, 7.834205709446429, 8.149076523982457, 8.948735159406311, 9.381025951525982, 9.867637514837967, 10.27997327443798, 10.92579679147785, 11.36546663542228, 11.63639696953220, 12.07348564695303, 12.73127484924863, 13.14228351408150
