| L(s) = 1 | + 2-s + 3-s + 4-s − 5-s + 6-s + 7-s + 8-s + 9-s − 10-s + 5·11-s + 12-s + 13-s + 14-s − 15-s + 16-s + 4·17-s + 18-s + 6·19-s − 20-s + 21-s + 5·22-s − 4·23-s + 24-s − 4·25-s + 26-s + 27-s + 28-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 0.577·3-s + 1/2·4-s − 0.447·5-s + 0.408·6-s + 0.377·7-s + 0.353·8-s + 1/3·9-s − 0.316·10-s + 1.50·11-s + 0.288·12-s + 0.277·13-s + 0.267·14-s − 0.258·15-s + 1/4·16-s + 0.970·17-s + 0.235·18-s + 1.37·19-s − 0.223·20-s + 0.218·21-s + 1.06·22-s − 0.834·23-s + 0.204·24-s − 4/5·25-s + 0.196·26-s + 0.192·27-s + 0.188·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 101478 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 101478 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(7.787765518\) |
| \(L(\frac12)\) |
\(\approx\) |
\(7.787765518\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 - T \) |
| 13 | \( 1 - T \) |
| 1301 | \( 1 + T \) |
| good | 5 | \( 1 + T + p T^{2} \) |
| 7 | \( 1 - T + p T^{2} \) |
| 11 | \( 1 - 5 T + p T^{2} \) |
| 17 | \( 1 - 4 T + p T^{2} \) |
| 19 | \( 1 - 6 T + p T^{2} \) |
| 23 | \( 1 + 4 T + p T^{2} \) |
| 29 | \( 1 - 2 T + p T^{2} \) |
| 31 | \( 1 - 4 T + p T^{2} \) |
| 37 | \( 1 - 10 T + p T^{2} \) |
| 41 | \( 1 + p T^{2} \) |
| 43 | \( 1 - 9 T + p T^{2} \) |
| 47 | \( 1 + 8 T + p T^{2} \) |
| 53 | \( 1 - 12 T + p T^{2} \) |
| 59 | \( 1 + 3 T + p T^{2} \) |
| 61 | \( 1 + T + p T^{2} \) |
| 67 | \( 1 + 9 T + p T^{2} \) |
| 71 | \( 1 + 12 T + p T^{2} \) |
| 73 | \( 1 + 3 T + p T^{2} \) |
| 79 | \( 1 - 10 T + p T^{2} \) |
| 83 | \( 1 + p T^{2} \) |
| 89 | \( 1 - 13 T + p T^{2} \) |
| 97 | \( 1 - 14 T + p T^{2} \) |
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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
−13.93565591813212, −13.35660491371564, −12.84020312274945, −12.07458150077273, −11.82150625612757, −11.62939075381474, −10.98325180808817, −10.15776493100237, −9.878291137912272, −9.251745684531150, −8.815883696011711, −8.039155207685226, −7.689514325713134, −7.391893314310359, −6.536510543683336, −6.109312328691813, −5.647661108340382, −4.828643943492185, −4.320678538423420, −3.869064529909563, −3.358908373987246, −2.838070548193712, −2.006749340282803, −1.311914208337447, −0.8259169905679432,
0.8259169905679432, 1.311914208337447, 2.006749340282803, 2.838070548193712, 3.358908373987246, 3.869064529909563, 4.320678538423420, 4.828643943492185, 5.647661108340382, 6.109312328691813, 6.536510543683336, 7.391893314310359, 7.689514325713134, 8.039155207685226, 8.815883696011711, 9.251745684531150, 9.878291137912272, 10.15776493100237, 10.98325180808817, 11.62939075381474, 11.82150625612757, 12.07458150077273, 12.84020312274945, 13.35660491371564, 13.93565591813212
