L(s) = 1 | − 3-s − 5-s + 7-s + 9-s − 11-s − 13-s + 15-s − 17-s + 19-s − 21-s − 23-s + 25-s − 27-s + 29-s + 31-s + 33-s − 35-s − 37-s + 39-s − 41-s + 43-s − 45-s − 47-s + 49-s + 51-s − 53-s + 55-s + ⋯ |
L(s) = 1 | − 3-s − 5-s + 7-s + 9-s − 11-s − 13-s + 15-s − 17-s + 19-s − 21-s − 23-s + 25-s − 27-s + 29-s + 31-s + 33-s − 35-s − 37-s + 39-s − 41-s + 43-s − 45-s − 47-s + 49-s + 51-s − 53-s + 55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6008 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6008 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.6054506552\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.6054506552\) |
\(L(1)\) |
\(\approx\) |
\(0.5986867926\) |
\(L(1)\) |
\(\approx\) |
\(0.5986867926\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 751 | \( 1 \) |
good | 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
−17.74087144377832071370418259316, −17.21737514968876218175367148321, −16.26060095330088942663370424347, −15.75797005072525811018506578827, −15.38687213991980621977406952951, −14.52595475184852303188444988043, −13.76183790211029461478840878254, −12.97722202069992115025645262920, −12.0801885090401359877133065840, −11.874746524216999323427765733424, −11.24954054385810922570899357408, −10.43368281596142530811462941931, −10.11918847445102456598952911899, −8.962510510734627648647583950029, −8.11936879935562303042151099816, −7.62347513936349963617134136197, −7.06441438380083275846675819705, −6.21316631087246457435148013816, −5.26634053706341457975354416478, −4.72423058287424696690614692347, −4.41327550175819110337593427809, −3.282535387168584963575090330512, −2.368173585512745134146173015075, −1.45943981696479553240765700960, −0.418601390892146812132739776437,
0.418601390892146812132739776437, 1.45943981696479553240765700960, 2.368173585512745134146173015075, 3.282535387168584963575090330512, 4.41327550175819110337593427809, 4.72423058287424696690614692347, 5.26634053706341457975354416478, 6.21316631087246457435148013816, 7.06441438380083275846675819705, 7.62347513936349963617134136197, 8.11936879935562303042151099816, 8.962510510734627648647583950029, 10.11918847445102456598952911899, 10.43368281596142530811462941931, 11.24954054385810922570899357408, 11.874746524216999323427765733424, 12.0801885090401359877133065840, 12.97722202069992115025645262920, 13.76183790211029461478840878254, 14.52595475184852303188444988043, 15.38687213991980621977406952951, 15.75797005072525811018506578827, 16.26060095330088942663370424347, 17.21737514968876218175367148321, 17.74087144377832071370418259316