| L(s) = 1 | + 3-s − 3.46·5-s − 7-s + 9-s + 5.46·11-s + 2·13-s − 3.46·15-s + 7.46·17-s − 6.92·19-s − 21-s + 5.46·23-s + 6.99·25-s + 27-s + 8.92·29-s + 2.92·31-s + 5.46·33-s + 3.46·35-s − 2·37-s + 2·39-s + 4.53·41-s − 8·43-s − 3.46·45-s − 2.92·47-s + 49-s + 7.46·51-s − 2·53-s − 18.9·55-s + ⋯ |
| L(s) = 1 | + 0.577·3-s − 1.54·5-s − 0.377·7-s + 0.333·9-s + 1.64·11-s + 0.554·13-s − 0.894·15-s + 1.81·17-s − 1.58·19-s − 0.218·21-s + 1.13·23-s + 1.39·25-s + 0.192·27-s + 1.65·29-s + 0.525·31-s + 0.951·33-s + 0.585·35-s − 0.328·37-s + 0.320·39-s + 0.708·41-s − 1.21·43-s − 0.516·45-s − 0.427·47-s + 0.142·49-s + 1.04·51-s − 0.274·53-s − 2.55·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 672 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 672 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.532459046\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.532459046\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 - T \) |
| 7 | \( 1 + T \) |
| good | 5 | \( 1 + 3.46T + 5T^{2} \) |
| 11 | \( 1 - 5.46T + 11T^{2} \) |
| 13 | \( 1 - 2T + 13T^{2} \) |
| 17 | \( 1 - 7.46T + 17T^{2} \) |
| 19 | \( 1 + 6.92T + 19T^{2} \) |
| 23 | \( 1 - 5.46T + 23T^{2} \) |
| 29 | \( 1 - 8.92T + 29T^{2} \) |
| 31 | \( 1 - 2.92T + 31T^{2} \) |
| 37 | \( 1 + 2T + 37T^{2} \) |
| 41 | \( 1 - 4.53T + 41T^{2} \) |
| 43 | \( 1 + 8T + 43T^{2} \) |
| 47 | \( 1 + 2.92T + 47T^{2} \) |
| 53 | \( 1 + 2T + 53T^{2} \) |
| 59 | \( 1 + 14.9T + 59T^{2} \) |
| 61 | \( 1 - 4.92T + 61T^{2} \) |
| 67 | \( 1 - 10.9T + 67T^{2} \) |
| 71 | \( 1 - 2.53T + 71T^{2} \) |
| 73 | \( 1 + 0.928T + 73T^{2} \) |
| 79 | \( 1 - 2.92T + 79T^{2} \) |
| 83 | \( 1 + 4T + 83T^{2} \) |
| 89 | \( 1 + 3.46T + 89T^{2} \) |
| 97 | \( 1 - 4.92T + 97T^{2} \) |
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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
−10.54037996881590863937881336647, −9.514613460313556899168862854738, −8.563282874025115237795605104710, −8.109239953343344007279947434768, −7.00469529776766762628040782257, −6.33507440054543428246941670987, −4.65005603433511873763283667591, −3.79438648256174728478462858442, −3.13650170611606548034931844134, −1.11697054605639928647811921199,
1.11697054605639928647811921199, 3.13650170611606548034931844134, 3.79438648256174728478462858442, 4.65005603433511873763283667591, 6.33507440054543428246941670987, 7.00469529776766762628040782257, 8.109239953343344007279947434768, 8.563282874025115237795605104710, 9.514613460313556899168862854738, 10.54037996881590863937881336647
