| L(s) = 1 | + 2-s + 4-s − 7-s + 8-s + 11-s + 13-s − 14-s + 16-s − 17-s + 19-s + 22-s − 23-s + 26-s − 28-s + 29-s + 32-s − 34-s + 37-s + 38-s − 41-s + 43-s + 44-s − 46-s + 47-s + 49-s + 52-s − 53-s + ⋯ |
| L(s) = 1 | + 2-s + 4-s − 7-s + 8-s + 11-s + 13-s − 14-s + 16-s − 17-s + 19-s + 22-s − 23-s + 26-s − 28-s + 29-s + 32-s − 34-s + 37-s + 38-s − 41-s + 43-s + 44-s − 46-s + 47-s + 49-s + 52-s − 53-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 465 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 465 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{1}{2})\) |
\(\approx\) |
\(2.566255542\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.566255542\) |
| \(L(1)\) |
\(\approx\) |
\(1.922734918\) |
| \(L(1)\) |
\(\approx\) |
\(1.922734918\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 5 | \( 1 \) |
| 31 | \( 1 \) |
| good | 2 | \( 1 + T \) |
| 7 | \( 1 - T \) |
| 11 | \( 1 + T \) |
| 13 | \( 1 + T \) |
| 17 | \( 1 - T \) |
| 19 | \( 1 + T \) |
| 23 | \( 1 - T \) |
| 29 | \( 1 + T \) |
| 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
−23.76466814019685310275836287717, −22.905213061826953571715313588439, −22.22515428520242197826609395715, −21.67682071577608301715105608558, −20.34667800700476427507987144029, −19.95943891108240007615214598439, −19.002760458476356555799148084527, −17.837815235110421064015403217329, −16.66726735915552615088524076675, −15.94230211292583209376329719327, −15.328888162918914533186698814288, −14.01931147362886434947686845056, −13.61241727509667371897897191383, −12.56118037871161733043693588135, −11.799912242567883831863128890586, −10.89147969011090576477938934440, −9.82333010883848616332619197769, −8.78096573226828567697608619437, −7.43714287661517742875840285683, −6.39765053516268174008922031977, −5.96934378272333035226620315305, −4.493290605327894532653040957459, −3.68870772821475805289630733491, −2.74587555784125290311133608034, −1.34858219736830140713188940177,
1.34858219736830140713188940177, 2.74587555784125290311133608034, 3.68870772821475805289630733491, 4.493290605327894532653040957459, 5.96934378272333035226620315305, 6.39765053516268174008922031977, 7.43714287661517742875840285683, 8.78096573226828567697608619437, 9.82333010883848616332619197769, 10.89147969011090576477938934440, 11.799912242567883831863128890586, 12.56118037871161733043693588135, 13.61241727509667371897897191383, 14.01931147362886434947686845056, 15.328888162918914533186698814288, 15.94230211292583209376329719327, 16.66726735915552615088524076675, 17.837815235110421064015403217329, 19.002760458476356555799148084527, 19.95943891108240007615214598439, 20.34667800700476427507987144029, 21.67682071577608301715105608558, 22.22515428520242197826609395715, 22.905213061826953571715313588439, 23.76466814019685310275836287717
