| L(s) = 1 | + 2-s + 3-s + 4-s + 2.80·5-s + 6-s + 8-s + 9-s + 2.80·10-s − 11-s + 12-s + 2.80·15-s + 16-s − 7.96·17-s + 18-s + 6.48·19-s + 2.80·20-s − 22-s − 5.28·23-s + 24-s + 2.87·25-s + 27-s + 5.12·29-s + 2.80·30-s + 8.96·31-s + 32-s − 33-s − 7.96·34-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 0.577·3-s + 0.5·4-s + 1.25·5-s + 0.408·6-s + 0.353·8-s + 0.333·9-s + 0.887·10-s − 0.301·11-s + 0.288·12-s + 0.724·15-s + 0.250·16-s − 1.93·17-s + 0.235·18-s + 1.48·19-s + 0.627·20-s − 0.213·22-s − 1.10·23-s + 0.204·24-s + 0.574·25-s + 0.192·27-s + 0.952·29-s + 0.512·30-s + 1.61·31-s + 0.176·32-s − 0.174·33-s − 1.36·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3234 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3234 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(4.684799123\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.684799123\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 - T \) |
| 7 | \( 1 \) |
| 11 | \( 1 + T \) |
| good | 5 | \( 1 - 2.80T + 5T^{2} \) |
| 13 | \( 1 + 13T^{2} \) |
| 17 | \( 1 + 7.96T + 17T^{2} \) |
| 19 | \( 1 - 6.48T + 19T^{2} \) |
| 23 | \( 1 + 5.28T + 23T^{2} \) |
| 29 | \( 1 - 5.12T + 29T^{2} \) |
| 31 | \( 1 - 8.96T + 31T^{2} \) |
| 37 | \( 1 - 10.4T + 37T^{2} \) |
| 41 | \( 1 - 5.09T + 41T^{2} \) |
| 43 | \( 1 - 11.4T + 43T^{2} \) |
| 47 | \( 1 + 0.322T + 47T^{2} \) |
| 53 | \( 1 - 8T + 53T^{2} \) |
| 59 | \( 1 - 0.871T + 59T^{2} \) |
| 61 | \( 1 + 2.15T + 61T^{2} \) |
| 67 | \( 1 + 7.09T + 67T^{2} \) |
| 71 | \( 1 + 7.44T + 71T^{2} \) |
| 73 | \( 1 + 12.5T + 73T^{2} \) |
| 79 | \( 1 - 6.80T + 79T^{2} \) |
| 83 | \( 1 + 15.0T + 83T^{2} \) |
| 89 | \( 1 + 1.61T + 89T^{2} \) |
| 97 | \( 1 + 7T + 97T^{2} \) |
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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
−8.708043564663346712492101037959, −7.84467749976348884436416024841, −7.05960484512122367580274885733, −6.17172279599073096756715180624, −5.77863900607316860031675382520, −4.66242801711219815870275478686, −4.14477436934902933626891039353, −2.64117314516565810080346788885, −2.53341879180570981254080568105, −1.24953794370594853295520798746,
1.24953794370594853295520798746, 2.53341879180570981254080568105, 2.64117314516565810080346788885, 4.14477436934902933626891039353, 4.66242801711219815870275478686, 5.77863900607316860031675382520, 6.17172279599073096756715180624, 7.05960484512122367580274885733, 7.84467749976348884436416024841, 8.708043564663346712492101037959
