| L(s) = 1 | + 1.16·2-s − 3-s − 0.649·4-s − 1.16·6-s − 4.28·7-s − 3.07·8-s + 9-s + 0.649·12-s + 5.16·13-s − 4.98·14-s − 2.27·16-s + 5·17-s + 1.16·18-s − 5.59·19-s + 4.28·21-s + 0.219·23-s + 3.07·24-s + 6.00·26-s − 27-s + 2.78·28-s − 6.41·29-s − 2.83·31-s + 3.50·32-s + 5.81·34-s − 0.649·36-s − 3.92·37-s − 6.50·38-s + ⋯ |
| L(s) = 1 | + 0.821·2-s − 0.577·3-s − 0.324·4-s − 0.474·6-s − 1.62·7-s − 1.08·8-s + 0.333·9-s + 0.187·12-s + 1.43·13-s − 1.33·14-s − 0.569·16-s + 1.21·17-s + 0.273·18-s − 1.28·19-s + 0.935·21-s + 0.0458·23-s + 0.628·24-s + 1.17·26-s − 0.192·27-s + 0.526·28-s − 1.19·29-s − 0.508·31-s + 0.620·32-s + 0.996·34-s − 0.108·36-s − 0.645·37-s − 1.05·38-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 9075 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 9075 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.9418364858\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.9418364858\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + T \) |
| 5 | \( 1 \) |
| 11 | \( 1 \) |
| good | 2 | \( 1 - 1.16T + 2T^{2} \) |
| 7 | \( 1 + 4.28T + 7T^{2} \) |
| 13 | \( 1 - 5.16T + 13T^{2} \) |
| 17 | \( 1 - 5T + 17T^{2} \) |
| 19 | \( 1 + 5.59T + 19T^{2} \) |
| 23 | \( 1 - 0.219T + 23T^{2} \) |
| 29 | \( 1 + 6.41T + 29T^{2} \) |
| 31 | \( 1 + 2.83T + 31T^{2} \) |
| 37 | \( 1 + 3.92T + 37T^{2} \) |
| 41 | \( 1 + 5.86T + 41T^{2} \) |
| 43 | \( 1 + 8.90T + 43T^{2} \) |
| 47 | \( 1 - 0.237T + 47T^{2} \) |
| 53 | \( 1 + 2.53T + 53T^{2} \) |
| 59 | \( 1 + 7.87T + 59T^{2} \) |
| 61 | \( 1 - 4.85T + 61T^{2} \) |
| 67 | \( 1 + 12.1T + 67T^{2} \) |
| 71 | \( 1 + 9.98T + 71T^{2} \) |
| 73 | \( 1 - 14.5T + 73T^{2} \) |
| 79 | \( 1 + 8.00T + 79T^{2} \) |
| 83 | \( 1 - 12.5T + 83T^{2} \) |
| 89 | \( 1 + 2.56T + 89T^{2} \) |
| 97 | \( 1 + 2.01T + 97T^{2} \) |
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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
−7.56493421042848222134498254588, −6.63048574652066797970690027846, −6.22994294021271135053870491280, −5.74394864444633166963372878390, −5.05919312088190404017509018071, −4.09783390031315695520047739430, −3.50283074185746436549402617688, −3.15564980172962449961296841848, −1.73572382515981140866181867344, −0.41904586877430747423577245681,
0.41904586877430747423577245681, 1.73572382515981140866181867344, 3.15564980172962449961296841848, 3.50283074185746436549402617688, 4.09783390031315695520047739430, 5.05919312088190404017509018071, 5.74394864444633166963372878390, 6.22994294021271135053870491280, 6.63048574652066797970690027846, 7.56493421042848222134498254588
