| L(s) = 1 | + 2-s + 3-s + 4-s + 6-s − 7-s + 8-s + 9-s + 11-s + 12-s − 14-s + 16-s + 17-s + 18-s + 19-s − 21-s + 22-s − 23-s + 24-s + 27-s − 28-s − 29-s − 31-s + 32-s + 33-s + 34-s + 36-s + 37-s + ⋯ |
| L(s) = 1 | + 2-s + 3-s + 4-s + 6-s − 7-s + 8-s + 9-s + 11-s + 12-s − 14-s + 16-s + 17-s + 18-s + 19-s − 21-s + 22-s − 23-s + 24-s + 27-s − 28-s − 29-s − 31-s + 32-s + 33-s + 34-s + 36-s + 37-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2665 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2665 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{1}{2})\) |
\(\approx\) |
\(5.213271078\) |
| \(L(\frac12)\) |
\(\approx\) |
\(5.213271078\) |
| \(L(1)\) |
\(\approx\) |
\(2.807782134\) |
| \(L(1)\) |
\(\approx\) |
\(2.807782134\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 \) |
| 13 | \( 1 \) |
| 41 | \( 1 \) |
| good | 2 | \( 1 + T \) |
| 3 | \( 1 + T \) |
| 7 | \( 1 - T \) |
| 11 | \( 1 + T \) |
| 17 | \( 1 + T \) |
| 19 | \( 1 + T \) |
| 23 | \( 1 - T \) |
| 29 | \( 1 - T \) |
| 31 | \( 1 - T \) |
| 37 | \( 1 + T \) |
| 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
−19.62624320569597819635813115609, −18.864642366115203633257269059248, −18.12851969821581824959281845542, −16.65436635578731403910606157637, −16.45083330469701861113693169603, −15.626837718618955970666868639835, −14.794903618391448665830275482211, −14.40873775853149447603205410862, −13.59583900529167668608622164988, −13.11701101742649235941604399774, −12.27592646442402501501253521058, −11.78448063634680778154994993286, −10.69907317829476804886921880183, −9.666769722718803476073831953987, −9.51518518482533136615712376469, −8.24398769785795184352879356348, −7.4845685429360720567383252108, −6.844220952744762927257128148011, −6.05332017589671405172497143089, −5.24822182920595692137734534173, −4.06815007388915536710785227933, −3.60583651531770503223513158159, −3.02643409693924321087650925086, −2.03848111442326867039404777206, −1.19552428054049151639955602583,
1.19552428054049151639955602583, 2.03848111442326867039404777206, 3.02643409693924321087650925086, 3.60583651531770503223513158159, 4.06815007388915536710785227933, 5.24822182920595692137734534173, 6.05332017589671405172497143089, 6.844220952744762927257128148011, 7.4845685429360720567383252108, 8.24398769785795184352879356348, 9.51518518482533136615712376469, 9.666769722718803476073831953987, 10.69907317829476804886921880183, 11.78448063634680778154994993286, 12.27592646442402501501253521058, 13.11701101742649235941604399774, 13.59583900529167668608622164988, 14.40873775853149447603205410862, 14.794903618391448665830275482211, 15.626837718618955970666868639835, 16.45083330469701861113693169603, 16.65436635578731403910606157637, 18.12851969821581824959281845542, 18.864642366115203633257269059248, 19.62624320569597819635813115609
