| 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 − 32-s − 33-s − 34-s + 36-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 − 32-s − 33-s − 34-s + 36-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 155 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 155 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{1}{2})\) |
\(\approx\) |
\(1.584900920\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.584900920\) |
| \(L(1)\) |
\(\approx\) |
\(1.009355177\) |
| \(L(1)\) |
\(\approx\) |
\(1.009355177\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 \) |
| 31 | \( 1 \) |
| 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 | \( 1 - T \) |
| 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
−27.57602625148412345143509858336, −26.46170293366109851016633699806, −25.90914645302545186228197424435, −25.23030920258361367680561957091, −24.147388418017948037749063637402, −22.93936430633474560982256512179, −21.27494312645721887522837039517, −20.66181975367558572704749590762, −19.66075382732379847654921225961, −18.77127966492803594562341716376, −18.185411618648244014250077273635, −16.49122033838853308786260325688, −15.87842824708895914781748003328, −14.89589108099870722837478848279, −13.462936635701975357042288150909, −12.55917614029182329291419157508, −10.970948691897027992161162179051, −9.87823258258194833522833825732, −9.15847286051048666335766148179, −7.98888396247113992126741514328, −7.1568549001624265776650302336, −5.7540624275658825720526749644, −3.51167575581166513323502633495, −2.656709463029098820625784200044, −1.01013246125190857650108132905,
1.01013246125190857650108132905, 2.656709463029098820625784200044, 3.51167575581166513323502633495, 5.7540624275658825720526749644, 7.1568549001624265776650302336, 7.98888396247113992126741514328, 9.15847286051048666335766148179, 9.87823258258194833522833825732, 10.970948691897027992161162179051, 12.55917614029182329291419157508, 13.462936635701975357042288150909, 14.89589108099870722837478848279, 15.87842824708895914781748003328, 16.49122033838853308786260325688, 18.185411618648244014250077273635, 18.77127966492803594562341716376, 19.66075382732379847654921225961, 20.66181975367558572704749590762, 21.27494312645721887522837039517, 22.93936430633474560982256512179, 24.147388418017948037749063637402, 25.23030920258361367680561957091, 25.90914645302545186228197424435, 26.46170293366109851016633699806, 27.57602625148412345143509858336
