| L(s) = 1 | − 2-s + 3-s + 4-s + 5-s − 6-s + 7-s − 8-s + 9-s − 10-s − 11-s + 12-s − 13-s − 14-s + 15-s + 16-s − 17-s − 18-s − 19-s + 20-s + 21-s + 22-s − 23-s − 24-s + 25-s + 26-s + 27-s + 28-s + ⋯ |
| L(s) = 1 | − 2-s + 3-s + 4-s + 5-s − 6-s + 7-s − 8-s + 9-s − 10-s − 11-s + 12-s − 13-s − 14-s + 15-s + 16-s − 17-s − 18-s − 19-s + 20-s + 21-s + 22-s − 23-s − 24-s + 25-s + 26-s + 27-s + 28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 109 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 109 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{1}{2})\) |
\(\approx\) |
\(1.082723856\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.082723856\) |
| \(L(1)\) |
\(\approx\) |
\(1.065971594\) |
| \(L(1)\) |
\(\approx\) |
\(1.065971594\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 109 | \( 1 \) |
| good | 2 | \( 1 - T \) |
| 3 | \( 1 + T \) |
| 5 | \( 1 + T \) |
| 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
−29.520334129127749572228700591039, −28.468100014049631295586325894755, −27.250946188048183926429277020939, −26.42073652012041932592760969407, −25.6532374167638786184107420922, −24.62094559981503723196735847394, −24.05620088106863369012120888114, −21.70174154151323617755049014584, −21.04726246541128660109817539730, −20.18511415581379988910060181043, −19.05852759530961450421851146004, −17.96330258773340693026257382060, −17.34001842619958701081346518848, −15.77402440390292922155945024351, −14.7942035799309337848548913481, −13.74274328509748199700612864008, −12.384543388042899682635337654044, −10.67496691250965725064437841847, −9.91188377242061786161359468430, −8.694108458771540703129355424390, −7.90133547299128819712428491523, −6.59270246833948403444968225621, −4.8376697312165710941950310399, −2.57920099173329354155216040558, −1.87221145476100712711633579311,
1.87221145476100712711633579311, 2.57920099173329354155216040558, 4.8376697312165710941950310399, 6.59270246833948403444968225621, 7.90133547299128819712428491523, 8.694108458771540703129355424390, 9.91188377242061786161359468430, 10.67496691250965725064437841847, 12.384543388042899682635337654044, 13.74274328509748199700612864008, 14.7942035799309337848548913481, 15.77402440390292922155945024351, 17.34001842619958701081346518848, 17.96330258773340693026257382060, 19.05852759530961450421851146004, 20.18511415581379988910060181043, 21.04726246541128660109817539730, 21.70174154151323617755049014584, 24.05620088106863369012120888114, 24.62094559981503723196735847394, 25.6532374167638786184107420922, 26.42073652012041932592760969407, 27.250946188048183926429277020939, 28.468100014049631295586325894755, 29.520334129127749572228700591039
