| L(s) = 1 | + 2-s − 3-s + 4-s − 6-s + 7-s + 8-s + 9-s + 11-s − 12-s + 14-s + 16-s − 17-s + 18-s + 19-s − 21-s + 22-s + 23-s − 24-s − 27-s + 28-s − 29-s + 31-s + 32-s − 33-s − 34-s + 36-s + 37-s + ⋯ |
| L(s) = 1 | + 2-s − 3-s + 4-s − 6-s + 7-s + 8-s + 9-s + 11-s − 12-s + 14-s + 16-s − 17-s + 18-s + 19-s − 21-s + 22-s + 23-s − 24-s − 27-s + 28-s − 29-s + 31-s + 32-s − 33-s − 34-s + 36-s + 37-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3055 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3055 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{1}{2})\) |
\(\approx\) |
\(5.405682939\) |
| \(L(\frac12)\) |
\(\approx\) |
\(5.405682939\) |
| \(L(1)\) |
\(\approx\) |
\(2.046193808\) |
| \(L(1)\) |
\(\approx\) |
\(2.046193808\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 \) |
| 13 | \( 1 \) |
| 47 | \( 1 \) |
| good | 2 | \( 1 + T \) |
| 3 | \( 1 - T \) |
| 7 | \( 1 + T \) |
| 11 | \( 1 + T \) |
| 17 | \( 1 - T \) |
| 19 | \( 1 + T \) |
| 23 | \( 1 + T \) |
| 29 | \( 1 - T \) |
| 31 | \( 1 + T \) |
| 37 | \( 1 + T \) |
| 41 | \( 1 + T \) |
| 43 | \( 1 + T \) |
| 53 | \( 1 - T \) |
| 59 | \( 1 - T \) |
| 61 | \( 1 + T \) |
| 67 | \( 1 - T \) |
| 71 | \( 1 - T \) |
| 73 | \( 1 - T \) |
| 79 | \( 1 + T \) |
| 83 | \( 1 + T \) |
| 89 | \( 1 - T \) |
| 97 | \( 1 + T \) |
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\(L(s) = \displaystyle\prod_p \ (1 - \alpha_{p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−18.906902348196423035207052630473, −17.88701349310097987137961811494, −17.40321847687452900959881628945, −16.7309192561194287743085288717, −16.02254890142297279883554609745, −15.273053974612936048662259333, −14.68736029125190925562877446923, −13.909371972620745304495730821646, −13.20370250591532659398334309832, −12.44747305673645178138308721918, −11.712004237943416128537200234856, −11.24412802345727299336170834663, −10.84018048268018610330199499228, −9.75714688256041643576768757193, −8.8966422353999008549150548273, −7.62732583410141410003710363738, −7.21699231483412505661996512593, −6.2601976508658625952099119885, −5.78683005861983510338183367233, −4.74636516756582074257946498767, −4.52669372842056106317286212457, −3.59130762737767981594425353690, −2.44186615167994081252149781540, −1.48327446464711611399734743436, −0.86574467853013888957712462667,
0.86574467853013888957712462667, 1.48327446464711611399734743436, 2.44186615167994081252149781540, 3.59130762737767981594425353690, 4.52669372842056106317286212457, 4.74636516756582074257946498767, 5.78683005861983510338183367233, 6.2601976508658625952099119885, 7.21699231483412505661996512593, 7.62732583410141410003710363738, 8.8966422353999008549150548273, 9.75714688256041643576768757193, 10.84018048268018610330199499228, 11.24412802345727299336170834663, 11.712004237943416128537200234856, 12.44747305673645178138308721918, 13.20370250591532659398334309832, 13.909371972620745304495730821646, 14.68736029125190925562877446923, 15.273053974612936048662259333, 16.02254890142297279883554609745, 16.7309192561194287743085288717, 17.40321847687452900959881628945, 17.88701349310097987137961811494, 18.906902348196423035207052630473
