| 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 + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 107 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 107 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{1}{2})\) |
\(\approx\) |
\(1.326852223\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.326852223\) |
| \(L(1)\) |
\(\approx\) |
\(0.9111276755\) |
| \(L(1)\) |
\(\approx\) |
\(0.9111276755\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 107 | \( 1 \) |
| 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
−29.2507140111264096034499664478, −28.14960124760015303204373695397, −27.04871897087953847297680943008, −26.49471452647039048371369461478, −25.39703258965952156230327048379, −24.680981528594938389175505521, −23.39650356421593954043645065110, −21.96844673492842249095618647735, −20.44994622216287872583535436397, −19.81318600232194171014385393698, −19.12980309394917793974016224391, −18.148053187499111510642509674952, −16.44276709701150967191770631156, −15.77499870531388299215492066204, −14.82590541909745539391871807166, −13.26211254951704896426933380367, −11.97180264986610149206081297316, −10.75367821674543183729850651015, −9.304695466462708620345463212961, −8.72812555651843937650658447358, −7.41713536583272743899328295646, −6.52750358073905121597717020075, −3.9027623586238545483928887445, −2.90812216403723678026329739840, −1.01532045680624064311140232848,
1.01532045680624064311140232848, 2.90812216403723678026329739840, 3.9027623586238545483928887445, 6.52750358073905121597717020075, 7.41713536583272743899328295646, 8.72812555651843937650658447358, 9.304695466462708620345463212961, 10.75367821674543183729850651015, 11.97180264986610149206081297316, 13.26211254951704896426933380367, 14.82590541909745539391871807166, 15.77499870531388299215492066204, 16.44276709701150967191770631156, 18.148053187499111510642509674952, 19.12980309394917793974016224391, 19.81318600232194171014385393698, 20.44994622216287872583535436397, 21.96844673492842249095618647735, 23.39650356421593954043645065110, 24.680981528594938389175505521, 25.39703258965952156230327048379, 26.49471452647039048371369461478, 27.04871897087953847297680943008, 28.14960124760015303204373695397, 29.2507140111264096034499664478
