L(s) = 1 | − 0.445·2-s + 1.24·3-s − 0.801·4-s + 5-s − 0.554·6-s + 0.801·8-s + 0.554·9-s − 0.445·10-s + 11-s − 12-s − 1.80·13-s + 1.24·15-s + 0.445·16-s + 17-s − 0.246·18-s − 0.801·20-s − 0.445·22-s − 0.445·23-s + 24-s + 25-s + 0.801·26-s − 0.554·27-s − 0.445·29-s − 0.554·30-s − 32-s + 1.24·33-s − 0.445·34-s + ⋯ |
L(s) = 1 | − 0.445·2-s + 1.24·3-s − 0.801·4-s + 5-s − 0.554·6-s + 0.801·8-s + 0.554·9-s − 0.445·10-s + 11-s − 12-s − 1.80·13-s + 1.24·15-s + 0.445·16-s + 17-s − 0.246·18-s − 0.801·20-s − 0.445·22-s − 0.445·23-s + 24-s + 25-s + 0.801·26-s − 0.554·27-s − 0.445·29-s − 0.554·30-s − 32-s + 1.24·33-s − 0.445·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 935 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 935 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(1.164992402\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.164992402\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 5 | \( 1 - T \) |
| 11 | \( 1 - T \) |
| 17 | \( 1 - T \) |
good | 2 | \( 1 + 0.445T + T^{2} \) |
| 3 | \( 1 - 1.24T + T^{2} \) |
| 7 | \( 1 - T^{2} \) |
| 13 | \( 1 + 1.80T + T^{2} \) |
| 19 | \( 1 - T^{2} \) |
| 23 | \( 1 + 0.445T + T^{2} \) |
| 29 | \( 1 + 0.445T + T^{2} \) |
| 31 | \( 1 - T^{2} \) |
| 37 | \( 1 + 1.80T + T^{2} \) |
| 41 | \( 1 - 1.24T + T^{2} \) |
| 43 | \( 1 - 1.24T + T^{2} \) |
| 47 | \( 1 - T^{2} \) |
| 53 | \( 1 - T^{2} \) |
| 59 | \( 1 + 1.80T + T^{2} \) |
| 61 | \( 1 + 1.80T + T^{2} \) |
| 67 | \( 1 - T^{2} \) |
| 71 | \( 1 - T^{2} \) |
| 73 | \( 1 - T^{2} \) |
| 79 | \( 1 - 1.24T + T^{2} \) |
| 83 | \( 1 + 1.80T + T^{2} \) |
| 89 | \( 1 - 1.24T + T^{2} \) |
| 97 | \( 1 + 0.445T + T^{2} \) |
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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
−9.886209929027275548663875603237, −9.227033540404136947646353134140, −9.005299079415044631187476810403, −7.82456395076873583611551554664, −7.28950148382340167205620306789, −5.89732131835972839432258383512, −4.92809450221644503163163825020, −3.86372449272077862085599800008, −2.72265882511049694531021323304, −1.61742562917136204123213401969,
1.61742562917136204123213401969, 2.72265882511049694531021323304, 3.86372449272077862085599800008, 4.92809450221644503163163825020, 5.89732131835972839432258383512, 7.28950148382340167205620306789, 7.82456395076873583611551554664, 9.005299079415044631187476810403, 9.227033540404136947646353134140, 9.886209929027275548663875603237