L(s) = 1 | − 2-s + 4-s − 0.939·5-s + 3.42·7-s − 8-s + 0.939·10-s − 0.393·11-s + 1.28·13-s − 3.42·14-s + 16-s − 7.81·17-s + 0.720·19-s − 0.939·20-s + 0.393·22-s − 0.995·23-s − 4.11·25-s − 1.28·26-s + 3.42·28-s − 0.330·29-s − 5.48·31-s − 32-s + 7.81·34-s − 3.21·35-s − 1.27·37-s − 0.720·38-s + 0.939·40-s − 12.4·41-s + ⋯ |
L(s) = 1 | − 0.707·2-s + 0.5·4-s − 0.420·5-s + 1.29·7-s − 0.353·8-s + 0.297·10-s − 0.118·11-s + 0.357·13-s − 0.914·14-s + 0.250·16-s − 1.89·17-s + 0.165·19-s − 0.210·20-s + 0.0839·22-s − 0.207·23-s − 0.823·25-s − 0.252·26-s + 0.646·28-s − 0.0614·29-s − 0.984·31-s − 0.176·32-s + 1.33·34-s − 0.543·35-s − 0.208·37-s − 0.116·38-s + 0.148·40-s − 1.93·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8046 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8046 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.219141848\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.219141848\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 \) |
| 149 | \( 1 - T \) |
good | 5 | \( 1 + 0.939T + 5T^{2} \) |
| 7 | \( 1 - 3.42T + 7T^{2} \) |
| 11 | \( 1 + 0.393T + 11T^{2} \) |
| 13 | \( 1 - 1.28T + 13T^{2} \) |
| 17 | \( 1 + 7.81T + 17T^{2} \) |
| 19 | \( 1 - 0.720T + 19T^{2} \) |
| 23 | \( 1 + 0.995T + 23T^{2} \) |
| 29 | \( 1 + 0.330T + 29T^{2} \) |
| 31 | \( 1 + 5.48T + 31T^{2} \) |
| 37 | \( 1 + 1.27T + 37T^{2} \) |
| 41 | \( 1 + 12.4T + 41T^{2} \) |
| 43 | \( 1 - 5.23T + 43T^{2} \) |
| 47 | \( 1 - 0.953T + 47T^{2} \) |
| 53 | \( 1 - 11.7T + 53T^{2} \) |
| 59 | \( 1 - 7.48T + 59T^{2} \) |
| 61 | \( 1 - 8.28T + 61T^{2} \) |
| 67 | \( 1 - 12.0T + 67T^{2} \) |
| 71 | \( 1 - 4.57T + 71T^{2} \) |
| 73 | \( 1 + 0.609T + 73T^{2} \) |
| 79 | \( 1 - 11.6T + 79T^{2} \) |
| 83 | \( 1 - 4.32T + 83T^{2} \) |
| 89 | \( 1 + 8.05T + 89T^{2} \) |
| 97 | \( 1 + 9.94T + 97T^{2} \) |
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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
−8.076251902112396602881112438595, −7.14949280001943543261424602774, −6.77871746778401229338703740974, −5.72898725958871683901417027358, −5.07655246674747864811678570740, −4.23243346397456432608254659242, −3.59367456425670868119626524991, −2.25248180780289274119503645808, −1.85345127829420384247578418232, −0.59865446866368116148643464740,
0.59865446866368116148643464740, 1.85345127829420384247578418232, 2.25248180780289274119503645808, 3.59367456425670868119626524991, 4.23243346397456432608254659242, 5.07655246674747864811678570740, 5.72898725958871683901417027358, 6.77871746778401229338703740974, 7.14949280001943543261424602774, 8.076251902112396602881112438595