L(s) = 1 | + 2.05·3-s − 2.46·5-s + 3.02·7-s + 1.21·9-s + 5.53·11-s + 0.663·13-s − 5.05·15-s + 5.11·17-s + 19-s + 6.21·21-s + 0.782·23-s + 1.06·25-s − 3.65·27-s + 7.26·29-s + 2.43·31-s + 11.3·33-s − 7.44·35-s + 10.9·37-s + 1.36·39-s − 4.18·41-s − 8.84·43-s − 3.00·45-s − 6.81·47-s + 2.15·49-s + 10.5·51-s + 0.456·53-s − 13.6·55-s + ⋯ |
L(s) = 1 | + 1.18·3-s − 1.10·5-s + 1.14·7-s + 0.406·9-s + 1.66·11-s + 0.184·13-s − 1.30·15-s + 1.24·17-s + 0.229·19-s + 1.35·21-s + 0.163·23-s + 0.212·25-s − 0.704·27-s + 1.34·29-s + 0.438·31-s + 1.97·33-s − 1.25·35-s + 1.80·37-s + 0.218·39-s − 0.653·41-s − 1.34·43-s − 0.447·45-s − 0.994·47-s + 0.308·49-s + 1.47·51-s + 0.0627·53-s − 1.83·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6004 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6004 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(3.489695537\) |
\(L(\frac12)\) |
\(\approx\) |
\(3.489695537\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 19 | \( 1 - T \) |
| 79 | \( 1 - T \) |
good | 3 | \( 1 - 2.05T + 3T^{2} \) |
| 5 | \( 1 + 2.46T + 5T^{2} \) |
| 7 | \( 1 - 3.02T + 7T^{2} \) |
| 11 | \( 1 - 5.53T + 11T^{2} \) |
| 13 | \( 1 - 0.663T + 13T^{2} \) |
| 17 | \( 1 - 5.11T + 17T^{2} \) |
| 23 | \( 1 - 0.782T + 23T^{2} \) |
| 29 | \( 1 - 7.26T + 29T^{2} \) |
| 31 | \( 1 - 2.43T + 31T^{2} \) |
| 37 | \( 1 - 10.9T + 37T^{2} \) |
| 41 | \( 1 + 4.18T + 41T^{2} \) |
| 43 | \( 1 + 8.84T + 43T^{2} \) |
| 47 | \( 1 + 6.81T + 47T^{2} \) |
| 53 | \( 1 - 0.456T + 53T^{2} \) |
| 59 | \( 1 + 13.8T + 59T^{2} \) |
| 61 | \( 1 + 10.9T + 61T^{2} \) |
| 67 | \( 1 - 2.39T + 67T^{2} \) |
| 71 | \( 1 - 10.3T + 71T^{2} \) |
| 73 | \( 1 - 12.5T + 73T^{2} \) |
| 83 | \( 1 - 13.3T + 83T^{2} \) |
| 89 | \( 1 + 14.4T + 89T^{2} \) |
| 97 | \( 1 - 4.05T + 97T^{2} \) |
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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
−8.194654082872027031080742786842, −7.72089286970607937095730993078, −6.85231733178982074597321157165, −6.06930172493927216259044917824, −4.89127761365522529253852194548, −4.31346093353919112441436442890, −3.53057344317050513885251738086, −3.05278470169080453386085020492, −1.78648010110422406916846113009, −1.01820513331213613625820984623,
1.01820513331213613625820984623, 1.78648010110422406916846113009, 3.05278470169080453386085020492, 3.53057344317050513885251738086, 4.31346093353919112441436442890, 4.89127761365522529253852194548, 6.06930172493927216259044917824, 6.85231733178982074597321157165, 7.72089286970607937095730993078, 8.194654082872027031080742786842