L(s) = 1 | + 0.547·3-s + 5-s + 7-s − 2.70·9-s − 11-s + 6.76·13-s + 0.547·15-s + 1.45·17-s + 5.30·19-s + 0.547·21-s + 2.51·23-s + 25-s − 3.11·27-s + 2·29-s − 6.24·31-s − 0.547·33-s + 35-s + 1.48·37-s + 3.70·39-s + 7.15·41-s + 1.60·43-s − 2.70·45-s − 3.66·47-s + 49-s + 0.795·51-s + 4.70·53-s − 55-s + ⋯ |
L(s) = 1 | + 0.315·3-s + 0.447·5-s + 0.377·7-s − 0.900·9-s − 0.301·11-s + 1.87·13-s + 0.141·15-s + 0.352·17-s + 1.21·19-s + 0.119·21-s + 0.524·23-s + 0.200·25-s − 0.600·27-s + 0.371·29-s − 1.12·31-s − 0.0952·33-s + 0.169·35-s + 0.244·37-s + 0.592·39-s + 1.11·41-s + 0.245·43-s − 0.402·45-s − 0.534·47-s + 0.142·49-s + 0.111·51-s + 0.646·53-s − 0.134·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6160 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6160 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.784210411\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.784210411\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 5 | \( 1 - T \) |
| 7 | \( 1 - T \) |
| 11 | \( 1 + T \) |
good | 3 | \( 1 - 0.547T + 3T^{2} \) |
| 13 | \( 1 - 6.76T + 13T^{2} \) |
| 17 | \( 1 - 1.45T + 17T^{2} \) |
| 19 | \( 1 - 5.30T + 19T^{2} \) |
| 23 | \( 1 - 2.51T + 23T^{2} \) |
| 29 | \( 1 - 2T + 29T^{2} \) |
| 31 | \( 1 + 6.24T + 31T^{2} \) |
| 37 | \( 1 - 1.48T + 37T^{2} \) |
| 41 | \( 1 - 7.15T + 41T^{2} \) |
| 43 | \( 1 - 1.60T + 43T^{2} \) |
| 47 | \( 1 + 3.66T + 47T^{2} \) |
| 53 | \( 1 - 4.70T + 53T^{2} \) |
| 59 | \( 1 + 7.34T + 59T^{2} \) |
| 61 | \( 1 + 5.55T + 61T^{2} \) |
| 67 | \( 1 - 0.391T + 67T^{2} \) |
| 71 | \( 1 + 12.4T + 71T^{2} \) |
| 73 | \( 1 - 6.54T + 73T^{2} \) |
| 79 | \( 1 + 12.0T + 79T^{2} \) |
| 83 | \( 1 - 15.6T + 83T^{2} \) |
| 89 | \( 1 + 5.18T + 89T^{2} \) |
| 97 | \( 1 + 6.21T + 97T^{2} \) |
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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
−8.028272469475406656636231029300, −7.55670656188751965422802257876, −6.55060511279694125685635499850, −5.74981687156121929725538737935, −5.47346421047166254754131141281, −4.40197609676264434402413910551, −3.41735495456790402077140998510, −2.93037725275282670420591751759, −1.79855385240207087440330348550, −0.904499328125522667844029867054,
0.904499328125522667844029867054, 1.79855385240207087440330348550, 2.93037725275282670420591751759, 3.41735495456790402077140998510, 4.40197609676264434402413910551, 5.47346421047166254754131141281, 5.74981687156121929725538737935, 6.55060511279694125685635499850, 7.55670656188751965422802257876, 8.028272469475406656636231029300