| L(s) = 1 | + 1.35·2-s − 0.169·4-s + 0.738·5-s + 3.44·7-s − 2.93·8-s + 0.998·10-s − 1.17·11-s − 2.89·13-s + 4.65·14-s − 3.63·16-s + 3.96·17-s − 3.02·19-s − 0.125·20-s − 1.58·22-s + 23-s − 4.45·25-s − 3.92·26-s − 0.583·28-s + 29-s + 10.3·31-s + 0.956·32-s + 5.36·34-s + 2.54·35-s + 3.14·37-s − 4.09·38-s − 2.16·40-s + 7.62·41-s + ⋯ |
| L(s) = 1 | + 0.956·2-s − 0.0848·4-s + 0.330·5-s + 1.30·7-s − 1.03·8-s + 0.315·10-s − 0.353·11-s − 0.803·13-s + 1.24·14-s − 0.908·16-s + 0.962·17-s − 0.694·19-s − 0.0279·20-s − 0.338·22-s + 0.208·23-s − 0.891·25-s − 0.769·26-s − 0.110·28-s + 0.185·29-s + 1.86·31-s + 0.169·32-s + 0.920·34-s + 0.429·35-s + 0.517·37-s − 0.664·38-s − 0.342·40-s + 1.19·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6003 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6003 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.298361203\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.298361203\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 23 | \( 1 - T \) |
| 29 | \( 1 - T \) |
| good | 2 | \( 1 - 1.35T + 2T^{2} \) |
| 5 | \( 1 - 0.738T + 5T^{2} \) |
| 7 | \( 1 - 3.44T + 7T^{2} \) |
| 11 | \( 1 + 1.17T + 11T^{2} \) |
| 13 | \( 1 + 2.89T + 13T^{2} \) |
| 17 | \( 1 - 3.96T + 17T^{2} \) |
| 19 | \( 1 + 3.02T + 19T^{2} \) |
| 31 | \( 1 - 10.3T + 31T^{2} \) |
| 37 | \( 1 - 3.14T + 37T^{2} \) |
| 41 | \( 1 - 7.62T + 41T^{2} \) |
| 43 | \( 1 - 6.17T + 43T^{2} \) |
| 47 | \( 1 + 11.0T + 47T^{2} \) |
| 53 | \( 1 - 12.0T + 53T^{2} \) |
| 59 | \( 1 + 1.75T + 59T^{2} \) |
| 61 | \( 1 - 2.90T + 61T^{2} \) |
| 67 | \( 1 - 8.65T + 67T^{2} \) |
| 71 | \( 1 - 8.41T + 71T^{2} \) |
| 73 | \( 1 - 1.23T + 73T^{2} \) |
| 79 | \( 1 + 15.7T + 79T^{2} \) |
| 83 | \( 1 - 9.53T + 83T^{2} \) |
| 89 | \( 1 - 10.0T + 89T^{2} \) |
| 97 | \( 1 + 12.9T + 97T^{2} \) |
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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
−8.084420873169014155597754229423, −7.42603823865171389159793570674, −6.38747784902117953067093072656, −5.74446285848768494869525437598, −5.06249524984590313965687093062, −4.59041882458181500928556894455, −3.88462469146394835946408020403, −2.79090036501926211796518112945, −2.14002989852133187741489858986, −0.835108296712397940733919997224,
0.835108296712397940733919997224, 2.14002989852133187741489858986, 2.79090036501926211796518112945, 3.88462469146394835946408020403, 4.59041882458181500928556894455, 5.06249524984590313965687093062, 5.74446285848768494869525437598, 6.38747784902117953067093072656, 7.42603823865171389159793570674, 8.084420873169014155597754229423
