| L(s) = 1 | + 2-s + 3-s + 4-s + 6-s + 7-s + 8-s + 9-s − 11-s + 12-s − 13-s + 14-s + 16-s − 17-s + 18-s + 19-s + 21-s − 22-s − 23-s + 24-s − 26-s + 27-s + 28-s − 29-s + 31-s + 32-s − 33-s − 34-s + ⋯ |
| L(s) = 1 | + 2-s + 3-s + 4-s + 6-s + 7-s + 8-s + 9-s − 11-s + 12-s − 13-s + 14-s + 16-s − 17-s + 18-s + 19-s + 21-s − 22-s − 23-s + 24-s − 26-s + 27-s + 28-s − 29-s + 31-s + 32-s − 33-s − 34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 505 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 505 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{1}{2})\) |
\(\approx\) |
\(3.735520152\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.735520152\) |
| \(L(1)\) |
\(\approx\) |
\(2.630430150\) |
| \(L(1)\) |
\(\approx\) |
\(2.630430150\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 \) |
| 101 | \( 1 \) |
| good | 2 | \( 1 + T \) |
| 3 | \( 1 + T \) |
| 7 | \( 1 + T \) |
| 11 | \( 1 - T \) |
| 13 | \( 1 - T \) |
| 17 | \( 1 - T \) |
| 19 | \( 1 + T \) |
| 23 | \( 1 - T \) |
| 29 | \( 1 - T \) |
| 31 | \( 1 + T \) |
| 37 | \( 1 - T \) |
| 41 | \( 1 - T \) |
| 43 | \( 1 - T \) |
| 47 | \( 1 - T \) |
| 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
−23.98153893571217014157756595198, −22.695025017420112025911170191472, −21.77298090219482548654581938750, −21.16025764916314515395736880595, −20.28926242319097914881887603374, −19.88735629222060423081068222409, −18.66471214223897306906818215458, −17.77282190148434621479072537535, −16.52933507792148936689400802216, −15.37333752630511182145891737311, −15.1411483524835834438258804969, −13.94867217906716390417141618112, −13.63251474649027800132674007869, −12.52295784040986455380210957890, −11.67456646713078841661439541550, −10.59744704258333341230898947672, −9.71795046947661985171453605049, −8.28205739700343247709205054787, −7.680369435819110289270186764422, −6.78837354798041101273825951783, −5.25464283552326886533718708755, −4.674805367346206483670537114365, −3.54043135825839309019116389478, −2.45953787029266063087822714454, −1.770591798051891954246472604872,
1.770591798051891954246472604872, 2.45953787029266063087822714454, 3.54043135825839309019116389478, 4.674805367346206483670537114365, 5.25464283552326886533718708755, 6.78837354798041101273825951783, 7.680369435819110289270186764422, 8.28205739700343247709205054787, 9.71795046947661985171453605049, 10.59744704258333341230898947672, 11.67456646713078841661439541550, 12.52295784040986455380210957890, 13.63251474649027800132674007869, 13.94867217906716390417141618112, 15.1411483524835834438258804969, 15.37333752630511182145891737311, 16.52933507792148936689400802216, 17.77282190148434621479072537535, 18.66471214223897306906818215458, 19.88735629222060423081068222409, 20.28926242319097914881887603374, 21.16025764916314515395736880595, 21.77298090219482548654581938750, 22.695025017420112025911170191472, 23.98153893571217014157756595198
