| L(s) = 1 | + 2-s − 3-s + 4-s − 5-s − 6-s + 8-s + 9-s − 10-s − 12-s + 6·13-s + 15-s + 16-s + 2·17-s + 18-s − 20-s − 23-s − 24-s + 25-s + 6·26-s − 27-s + 6·29-s + 30-s + 8·31-s + 32-s + 2·34-s + 36-s + 10·37-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 0.577·3-s + 1/2·4-s − 0.447·5-s − 0.408·6-s + 0.353·8-s + 1/3·9-s − 0.316·10-s − 0.288·12-s + 1.66·13-s + 0.258·15-s + 1/4·16-s + 0.485·17-s + 0.235·18-s − 0.223·20-s − 0.208·23-s − 0.204·24-s + 1/5·25-s + 1.17·26-s − 0.192·27-s + 1.11·29-s + 0.182·30-s + 1.43·31-s + 0.176·32-s + 0.342·34-s + 1/6·36-s + 1.64·37-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 690 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 690 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.949253368\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.949253368\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + T \) |
| 5 | \( 1 + T \) |
| 23 | \( 1 + T \) |
| good | 7 | \( 1 + p T^{2} \) |
| 11 | \( 1 + p T^{2} \) |
| 13 | \( 1 - 6 T + p T^{2} \) |
| 17 | \( 1 - 2 T + p T^{2} \) |
| 19 | \( 1 + p T^{2} \) |
| 29 | \( 1 - 6 T + p T^{2} \) |
| 31 | \( 1 - 8 T + p T^{2} \) |
| 37 | \( 1 - 10 T + p T^{2} \) |
| 41 | \( 1 + 6 T + p T^{2} \) |
| 43 | \( 1 + 8 T + p T^{2} \) |
| 47 | \( 1 - 8 T + p T^{2} \) |
| 53 | \( 1 + 6 T + p T^{2} \) |
| 59 | \( 1 + 4 T + p T^{2} \) |
| 61 | \( 1 + 6 T + p T^{2} \) |
| 67 | \( 1 - 8 T + p T^{2} \) |
| 71 | \( 1 + 8 T + p T^{2} \) |
| 73 | \( 1 - 10 T + p T^{2} \) |
| 79 | \( 1 + 8 T + p T^{2} \) |
| 83 | \( 1 + 8 T + p T^{2} \) |
| 89 | \( 1 + 6 T + p T^{2} \) |
| 97 | \( 1 - 18 T + p T^{2} \) |
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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
−10.69696234533884622394355749914, −9.871544993622520353059832892538, −8.537206037121947139884084053940, −7.82443879413786556719070782655, −6.56989379181934243398131311271, −6.08309634878621601794680317739, −4.93421749373981937940111618456, −4.04939689132892717692595875585, −3.02400108992575685817499466920, −1.22203178801211012941035581235,
1.22203178801211012941035581235, 3.02400108992575685817499466920, 4.04939689132892717692595875585, 4.93421749373981937940111618456, 6.08309634878621601794680317739, 6.56989379181934243398131311271, 7.82443879413786556719070782655, 8.537206037121947139884084053940, 9.871544993622520353059832892538, 10.69696234533884622394355749914
