| L(s) = 1 | − 7-s + 2·13-s − 6·17-s − 8·19-s + 6·29-s − 4·31-s − 10·37-s + 6·41-s − 4·43-s + 49-s + 6·53-s − 12·59-s + 10·61-s − 4·67-s − 12·71-s + 10·73-s + 8·79-s − 12·83-s + 6·89-s − 2·91-s + 10·97-s + 101-s + 103-s + 107-s + 109-s + 113-s + 6·119-s + ⋯ |
| L(s) = 1 | − 0.377·7-s + 0.554·13-s − 1.45·17-s − 1.83·19-s + 1.11·29-s − 0.718·31-s − 1.64·37-s + 0.937·41-s − 0.609·43-s + 1/7·49-s + 0.824·53-s − 1.56·59-s + 1.28·61-s − 0.488·67-s − 1.42·71-s + 1.17·73-s + 0.900·79-s − 1.31·83-s + 0.635·89-s − 0.209·91-s + 1.01·97-s + 0.0995·101-s + 0.0985·103-s + 0.0966·107-s + 0.0957·109-s + 0.0940·113-s + 0.550·119-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 100800 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 100800 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.7382571734\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.7382571734\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 5 | \( 1 \) |
| 7 | \( 1 + T \) |
| good | 11 | \( 1 + p T^{2} \) |
| 13 | \( 1 - 2 T + p T^{2} \) |
| 17 | \( 1 + 6 T + p T^{2} \) |
| 19 | \( 1 + 8 T + p T^{2} \) |
| 23 | \( 1 + p T^{2} \) |
| 29 | \( 1 - 6 T + p T^{2} \) |
| 31 | \( 1 + 4 T + p T^{2} \) |
| 37 | \( 1 + 10 T + p T^{2} \) |
| 41 | \( 1 - 6 T + p T^{2} \) |
| 43 | \( 1 + 4 T + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 - 6 T + p T^{2} \) |
| 59 | \( 1 + 12 T + p T^{2} \) |
| 61 | \( 1 - 10 T + p T^{2} \) |
| 67 | \( 1 + 4 T + p T^{2} \) |
| 71 | \( 1 + 12 T + p T^{2} \) |
| 73 | \( 1 - 10 T + p T^{2} \) |
| 79 | \( 1 - 8 T + p T^{2} \) |
| 83 | \( 1 + 12 T + p T^{2} \) |
| 89 | \( 1 - 6 T + p T^{2} \) |
| 97 | \( 1 - 10 T + p T^{2} \) |
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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
−13.52365077758934, −13.33292631325509, −12.84871461105401, −12.27908382353114, −11.90849072665885, −11.13976090269187, −10.75737019987026, −10.47705332700366, −9.859736102489920, −9.043929614193716, −8.874422679762254, −8.390191535490611, −7.819698099281401, −6.987018259424420, −6.681075200218618, −6.256112042153657, −5.662754139633853, −4.939121558756217, −4.378415451730351, −3.939010875340718, −3.302092102194378, −2.507333825325232, −2.068224631795731, −1.312405274790164, −0.2655213700936548,
0.2655213700936548, 1.312405274790164, 2.068224631795731, 2.507333825325232, 3.302092102194378, 3.939010875340718, 4.378415451730351, 4.939121558756217, 5.662754139633853, 6.256112042153657, 6.681075200218618, 6.987018259424420, 7.819698099281401, 8.390191535490611, 8.874422679762254, 9.043929614193716, 9.859736102489920, 10.47705332700366, 10.75737019987026, 11.13976090269187, 11.90849072665885, 12.27908382353114, 12.84871461105401, 13.33292631325509, 13.52365077758934
