| L(s) = 1 | + 1.01·3-s − 2.65·5-s − 2.23·7-s − 1.97·9-s − 5.92·13-s − 2.69·15-s − 5.68·17-s − 19-s − 2.27·21-s − 2.64·23-s + 2.04·25-s − 5.04·27-s − 3.34·29-s + 2.69·31-s + 5.94·35-s − 9.80·37-s − 6.00·39-s − 4.78·41-s − 7.53·43-s + 5.23·45-s − 3.29·47-s − 1.98·49-s − 5.76·51-s + 10.1·53-s − 1.01·57-s − 0.344·59-s + 11.9·61-s + ⋯ |
| L(s) = 1 | + 0.585·3-s − 1.18·5-s − 0.846·7-s − 0.657·9-s − 1.64·13-s − 0.695·15-s − 1.37·17-s − 0.229·19-s − 0.495·21-s − 0.552·23-s + 0.409·25-s − 0.970·27-s − 0.620·29-s + 0.483·31-s + 1.00·35-s − 1.61·37-s − 0.961·39-s − 0.747·41-s − 1.14·43-s + 0.780·45-s − 0.480·47-s − 0.283·49-s − 0.807·51-s + 1.39·53-s − 0.134·57-s − 0.0448·59-s + 1.53·61-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 9196 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 9196 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.04693958104\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.04693958104\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 11 | \( 1 \) |
| 19 | \( 1 + T \) |
| good | 3 | \( 1 - 1.01T + 3T^{2} \) |
| 5 | \( 1 + 2.65T + 5T^{2} \) |
| 7 | \( 1 + 2.23T + 7T^{2} \) |
| 13 | \( 1 + 5.92T + 13T^{2} \) |
| 17 | \( 1 + 5.68T + 17T^{2} \) |
| 23 | \( 1 + 2.64T + 23T^{2} \) |
| 29 | \( 1 + 3.34T + 29T^{2} \) |
| 31 | \( 1 - 2.69T + 31T^{2} \) |
| 37 | \( 1 + 9.80T + 37T^{2} \) |
| 41 | \( 1 + 4.78T + 41T^{2} \) |
| 43 | \( 1 + 7.53T + 43T^{2} \) |
| 47 | \( 1 + 3.29T + 47T^{2} \) |
| 53 | \( 1 - 10.1T + 53T^{2} \) |
| 59 | \( 1 + 0.344T + 59T^{2} \) |
| 61 | \( 1 - 11.9T + 61T^{2} \) |
| 67 | \( 1 - 7.69T + 67T^{2} \) |
| 71 | \( 1 - 5.04T + 71T^{2} \) |
| 73 | \( 1 - 7.39T + 73T^{2} \) |
| 79 | \( 1 + 11.7T + 79T^{2} \) |
| 83 | \( 1 + 3.39T + 83T^{2} \) |
| 89 | \( 1 + 5.43T + 89T^{2} \) |
| 97 | \( 1 - 1.50T + 97T^{2} \) |
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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
−7.81823216191494081552957899701, −6.87589245726792216706907661263, −6.80141122438723233051870151361, −5.56433428450513110156485209266, −4.88429817090732801077409826423, −4.00816971653703383012844887633, −3.49951808877423251395777730368, −2.66965583910068881545372819637, −2.04116351629535199924271759980, −0.091833382525184405788364714865,
0.091833382525184405788364714865, 2.04116351629535199924271759980, 2.66965583910068881545372819637, 3.49951808877423251395777730368, 4.00816971653703383012844887633, 4.88429817090732801077409826423, 5.56433428450513110156485209266, 6.80141122438723233051870151361, 6.87589245726792216706907661263, 7.81823216191494081552957899701
