| L(s) = 1 | − 1.24·2-s + 0.554·4-s − 7-s + 0.554·8-s + 9-s + 0.445·11-s + 1.24·14-s − 1.24·16-s − 1.24·18-s − 0.554·22-s − 1.80·23-s + 25-s − 0.554·28-s + 1.24·29-s + 0.999·32-s + 0.554·36-s + 1.80·37-s + 1.24·43-s + 0.246·44-s + 2.24·46-s + 49-s − 1.24·50-s − 1.80·53-s − 0.554·56-s − 1.55·58-s − 63-s + 1.80·67-s + ⋯ |
| L(s) = 1 | − 1.24·2-s + 0.554·4-s − 7-s + 0.554·8-s + 9-s + 0.445·11-s + 1.24·14-s − 1.24·16-s − 1.24·18-s − 0.554·22-s − 1.80·23-s + 25-s − 0.554·28-s + 1.24·29-s + 0.999·32-s + 0.554·36-s + 1.80·37-s + 1.24·43-s + 0.246·44-s + 2.24·46-s + 49-s − 1.24·50-s − 1.80·53-s − 0.554·56-s − 1.55·58-s − 63-s + 1.80·67-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1183 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1183 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{1}{2})\) |
\(\approx\) |
\(0.5321548633\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.5321548633\) |
| \(L(1)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 7 | \( 1 + T \) |
| 13 | \( 1 \) |
| good | 2 | \( 1 + 1.24T + T^{2} \) |
| 3 | \( 1 - T^{2} \) |
| 5 | \( 1 - T^{2} \) |
| 11 | \( 1 - 0.445T + T^{2} \) |
| 17 | \( 1 - T^{2} \) |
| 19 | \( 1 - T^{2} \) |
| 23 | \( 1 + 1.80T + T^{2} \) |
| 29 | \( 1 - 1.24T + T^{2} \) |
| 31 | \( 1 - T^{2} \) |
| 37 | \( 1 - 1.80T + T^{2} \) |
| 41 | \( 1 - T^{2} \) |
| 43 | \( 1 - 1.24T + T^{2} \) |
| 47 | \( 1 - T^{2} \) |
| 53 | \( 1 + 1.80T + T^{2} \) |
| 59 | \( 1 - T^{2} \) |
| 61 | \( 1 - T^{2} \) |
| 67 | \( 1 - 1.80T + T^{2} \) |
| 71 | \( 1 - 1.80T + T^{2} \) |
| 73 | \( 1 - T^{2} \) |
| 79 | \( 1 + 0.445T + T^{2} \) |
| 83 | \( 1 - T^{2} \) |
| 89 | \( 1 - T^{2} \) |
| 97 | \( 1 - T^{2} \) |
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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
−9.715099885926382065175078417499, −9.433356175904036541330751869312, −8.379813221388291916201479827835, −7.68879662274753246182889360450, −6.78470594796732293025529582559, −6.18789536554597377082779789863, −4.66269600646787505217708461679, −3.84507302280474419394923189885, −2.39852851364531292746680904046, −1.02384959906269910220620087662,
1.02384959906269910220620087662, 2.39852851364531292746680904046, 3.84507302280474419394923189885, 4.66269600646787505217708461679, 6.18789536554597377082779789863, 6.78470594796732293025529582559, 7.68879662274753246182889360450, 8.379813221388291916201479827835, 9.433356175904036541330751869312, 9.715099885926382065175078417499
