| L(s) = 1 | − 11-s − 8·19-s + 4·23-s − 5·25-s + 2·29-s + 4·31-s − 2·37-s + 8·41-s − 8·43-s + 12·47-s + 6·53-s + 4·59-s + 8·61-s + 8·67-s + 12·71-s − 4·73-s − 8·79-s − 12·89-s − 8·97-s + 101-s + 103-s + 107-s + 109-s + 113-s + ⋯ |
| L(s) = 1 | − 0.301·11-s − 1.83·19-s + 0.834·23-s − 25-s + 0.371·29-s + 0.718·31-s − 0.328·37-s + 1.24·41-s − 1.21·43-s + 1.75·47-s + 0.824·53-s + 0.520·59-s + 1.02·61-s + 0.977·67-s + 1.42·71-s − 0.468·73-s − 0.900·79-s − 1.27·89-s − 0.812·97-s + 0.0995·101-s + 0.0985·103-s + 0.0966·107-s + 0.0957·109-s + 0.0940·113-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 310464 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 310464 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.911034766\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.911034766\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 7 | \( 1 \) |
| 11 | \( 1 + T \) |
| good | 5 | \( 1 + p T^{2} \) |
| 13 | \( 1 + p T^{2} \) |
| 17 | \( 1 + p T^{2} \) |
| 19 | \( 1 + 8 T + p T^{2} \) |
| 23 | \( 1 - 4 T + p T^{2} \) |
| 29 | \( 1 - 2 T + p T^{2} \) |
| 31 | \( 1 - 4 T + p T^{2} \) |
| 37 | \( 1 + 2 T + p T^{2} \) |
| 41 | \( 1 - 8 T + p T^{2} \) |
| 43 | \( 1 + 8 T + p T^{2} \) |
| 47 | \( 1 - 12 T + p T^{2} \) |
| 53 | \( 1 - 6 T + p T^{2} \) |
| 59 | \( 1 - 4 T + p T^{2} \) |
| 61 | \( 1 - 8 T + p T^{2} \) |
| 67 | \( 1 - 8 T + p T^{2} \) |
| 71 | \( 1 - 12 T + p T^{2} \) |
| 73 | \( 1 + 4 T + p T^{2} \) |
| 79 | \( 1 + 8 T + p T^{2} \) |
| 83 | \( 1 + p T^{2} \) |
| 89 | \( 1 + 12 T + p T^{2} \) |
| 97 | \( 1 + 8 T + p T^{2} \) |
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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
−12.72084851794057, −12.23834713021613, −11.77490564546922, −11.12210784684185, −10.98075935674673, −10.28641210690441, −10.02680150818267, −9.502197850562970, −8.854965572846180, −8.463465966947845, −8.221859300184297, −7.476236698455420, −7.089533268278357, −6.555809051815048, −6.123934244548427, −5.582379089258955, −5.087873292939483, −4.515982328955071, −3.971273681184413, −3.677929445601411, −2.674296469926729, −2.484627074171846, −1.838475042114997, −1.043220243120622, −0.3985274329449458,
0.3985274329449458, 1.043220243120622, 1.838475042114997, 2.484627074171846, 2.674296469926729, 3.677929445601411, 3.971273681184413, 4.515982328955071, 5.087873292939483, 5.582379089258955, 6.123934244548427, 6.555809051815048, 7.089533268278357, 7.476236698455420, 8.221859300184297, 8.463465966947845, 8.854965572846180, 9.502197850562970, 10.02680150818267, 10.28641210690441, 10.98075935674673, 11.12210784684185, 11.77490564546922, 12.23834713021613, 12.72084851794057
