| L(s) = 1 | + 2.64·3-s + 3.64·5-s + 2.64·7-s + 4.00·9-s + 0.645·11-s + 0.354·13-s + 9.64·15-s + 3·17-s − 5.64·19-s + 7.00·21-s + 7.29·23-s + 8.29·25-s + 2.64·27-s + 3·29-s + 2.64·31-s + 1.70·33-s + 9.64·35-s + 2.29·37-s + 0.937·39-s − 12·41-s − 8·43-s + 14.5·45-s − 10.9·47-s + 7.93·51-s − 13.2·53-s + 2.35·55-s − 14.9·57-s + ⋯ |
| L(s) = 1 | + 1.52·3-s + 1.63·5-s + 0.999·7-s + 1.33·9-s + 0.194·11-s + 0.0982·13-s + 2.49·15-s + 0.727·17-s − 1.29·19-s + 1.52·21-s + 1.52·23-s + 1.65·25-s + 0.509·27-s + 0.557·29-s + 0.475·31-s + 0.297·33-s + 1.63·35-s + 0.376·37-s + 0.150·39-s − 1.87·41-s − 1.21·43-s + 2.17·45-s − 1.59·47-s + 1.11·51-s − 1.82·53-s + 0.317·55-s − 1.97·57-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5312 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5312 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(5.686656314\) |
| \(L(\frac12)\) |
\(\approx\) |
\(5.686656314\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 83 | \( 1 - T \) |
| good | 3 | \( 1 - 2.64T + 3T^{2} \) |
| 5 | \( 1 - 3.64T + 5T^{2} \) |
| 7 | \( 1 - 2.64T + 7T^{2} \) |
| 11 | \( 1 - 0.645T + 11T^{2} \) |
| 13 | \( 1 - 0.354T + 13T^{2} \) |
| 17 | \( 1 - 3T + 17T^{2} \) |
| 19 | \( 1 + 5.64T + 19T^{2} \) |
| 23 | \( 1 - 7.29T + 23T^{2} \) |
| 29 | \( 1 - 3T + 29T^{2} \) |
| 31 | \( 1 - 2.64T + 31T^{2} \) |
| 37 | \( 1 - 2.29T + 37T^{2} \) |
| 41 | \( 1 + 12T + 41T^{2} \) |
| 43 | \( 1 + 8T + 43T^{2} \) |
| 47 | \( 1 + 10.9T + 47T^{2} \) |
| 53 | \( 1 + 13.2T + 53T^{2} \) |
| 59 | \( 1 + 7.93T + 59T^{2} \) |
| 61 | \( 1 - T + 61T^{2} \) |
| 67 | \( 1 + 14T + 67T^{2} \) |
| 71 | \( 1 - 13.2T + 71T^{2} \) |
| 73 | \( 1 - 0.937T + 73T^{2} \) |
| 79 | \( 1 + 13.6T + 79T^{2} \) |
| 89 | \( 1 - 7.29T + 89T^{2} \) |
| 97 | \( 1 - 15.2T + 97T^{2} \) |
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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
−8.435718713651247675124525975860, −7.69846450083600906086720947761, −6.71888813288908334944673382569, −6.19606369257255430074396421563, −5.04557149358299706767790451289, −4.68450609219493970482392999684, −3.36590600488516705180080420970, −2.80176534501153861267536476703, −1.78293079408349821071583148555, −1.50804665773188310019596989090,
1.50804665773188310019596989090, 1.78293079408349821071583148555, 2.80176534501153861267536476703, 3.36590600488516705180080420970, 4.68450609219493970482392999684, 5.04557149358299706767790451289, 6.19606369257255430074396421563, 6.71888813288908334944673382569, 7.69846450083600906086720947761, 8.435718713651247675124525975860
