| L(s) = 1 | + 2-s + 4-s − 1.61·5-s + 8-s + 9-s − 1.61·10-s + 0.618·13-s + 16-s + 0.618·17-s + 18-s − 1.61·20-s + 1.61·25-s + 0.618·26-s + 0.618·29-s + 32-s + 0.618·34-s + 36-s − 1.61·37-s − 1.61·40-s − 1.61·41-s − 1.61·45-s + 49-s + 1.61·50-s + 0.618·52-s − 1.61·53-s + 0.618·58-s + 0.618·61-s + ⋯ |
| L(s) = 1 | + 2-s + 4-s − 1.61·5-s + 8-s + 9-s − 1.61·10-s + 0.618·13-s + 16-s + 0.618·17-s + 18-s − 1.61·20-s + 1.61·25-s + 0.618·26-s + 0.618·29-s + 32-s + 0.618·34-s + 36-s − 1.61·37-s − 1.61·40-s − 1.61·41-s − 1.61·45-s + 49-s + 1.61·50-s + 0.618·52-s − 1.61·53-s + 0.618·58-s + 0.618·61-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1444 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1444 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{1}{2})\) |
\(\approx\) |
\(1.773736260\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.773736260\) |
| \(L(1)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - T \) |
| 19 | \( 1 \) |
| good | 3 | \( 1 - T^{2} \) |
| 5 | \( 1 + 1.61T + T^{2} \) |
| 7 | \( 1 - T^{2} \) |
| 11 | \( 1 - T^{2} \) |
| 13 | \( 1 - 0.618T + T^{2} \) |
| 17 | \( 1 - 0.618T + T^{2} \) |
| 23 | \( 1 - T^{2} \) |
| 29 | \( 1 - 0.618T + T^{2} \) |
| 31 | \( 1 - T^{2} \) |
| 37 | \( 1 + 1.61T + T^{2} \) |
| 41 | \( 1 + 1.61T + T^{2} \) |
| 43 | \( 1 - T^{2} \) |
| 47 | \( 1 - T^{2} \) |
| 53 | \( 1 + 1.61T + T^{2} \) |
| 59 | \( 1 - T^{2} \) |
| 61 | \( 1 - 0.618T + T^{2} \) |
| 67 | \( 1 - T^{2} \) |
| 71 | \( 1 - T^{2} \) |
| 73 | \( 1 + 1.61T + T^{2} \) |
| 79 | \( 1 - T^{2} \) |
| 83 | \( 1 - T^{2} \) |
| 89 | \( 1 + 1.61T + T^{2} \) |
| 97 | \( 1 - 0.618T + T^{2} \) |
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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
−10.02849723603225236203874858098, −8.628030587435983077100898783984, −7.915566344163055930494806306469, −7.17660430637350958581965305651, −6.60528638854424029309717881312, −5.35761959889671297849470314880, −4.48732510412639402399370797454, −3.80420260796449060522666033193, −3.12200734144396943976128537623, −1.46113443310373748222405680555,
1.46113443310373748222405680555, 3.12200734144396943976128537623, 3.80420260796449060522666033193, 4.48732510412639402399370797454, 5.35761959889671297849470314880, 6.60528638854424029309717881312, 7.17660430637350958581965305651, 7.915566344163055930494806306469, 8.628030587435983077100898783984, 10.02849723603225236203874858098
