| L(s) = 1 | − 2-s − 0.252·3-s + 4-s + 2.45·5-s + 0.252·6-s + 7-s − 8-s − 2.93·9-s − 2.45·10-s + 5.92·11-s − 0.252·12-s − 1.62·13-s − 14-s − 0.620·15-s + 16-s + 5.77·17-s + 2.93·18-s + 6.28·19-s + 2.45·20-s − 0.252·21-s − 5.92·22-s − 3.10·23-s + 0.252·24-s + 1.04·25-s + 1.62·26-s + 1.49·27-s + 28-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 0.145·3-s + 0.5·4-s + 1.09·5-s + 0.103·6-s + 0.377·7-s − 0.353·8-s − 0.978·9-s − 0.777·10-s + 1.78·11-s − 0.0728·12-s − 0.450·13-s − 0.267·14-s − 0.160·15-s + 0.250·16-s + 1.40·17-s + 0.692·18-s + 1.44·19-s + 0.549·20-s − 0.0550·21-s − 1.26·22-s − 0.647·23-s + 0.0515·24-s + 0.209·25-s + 0.318·26-s + 0.288·27-s + 0.188·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6034 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6034 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.116295928\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.116295928\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + T \) |
| 7 | \( 1 - T \) |
| 431 | \( 1 + T \) |
| good | 3 | \( 1 + 0.252T + 3T^{2} \) |
| 5 | \( 1 - 2.45T + 5T^{2} \) |
| 11 | \( 1 - 5.92T + 11T^{2} \) |
| 13 | \( 1 + 1.62T + 13T^{2} \) |
| 17 | \( 1 - 5.77T + 17T^{2} \) |
| 19 | \( 1 - 6.28T + 19T^{2} \) |
| 23 | \( 1 + 3.10T + 23T^{2} \) |
| 29 | \( 1 - 5.44T + 29T^{2} \) |
| 31 | \( 1 - 7.32T + 31T^{2} \) |
| 37 | \( 1 - 1.65T + 37T^{2} \) |
| 41 | \( 1 - 5.03T + 41T^{2} \) |
| 43 | \( 1 - 3.50T + 43T^{2} \) |
| 47 | \( 1 + 7.09T + 47T^{2} \) |
| 53 | \( 1 + 7.18T + 53T^{2} \) |
| 59 | \( 1 + 9.12T + 59T^{2} \) |
| 61 | \( 1 - 6.32T + 61T^{2} \) |
| 67 | \( 1 + 8.09T + 67T^{2} \) |
| 71 | \( 1 + 1.75T + 71T^{2} \) |
| 73 | \( 1 + 10.8T + 73T^{2} \) |
| 79 | \( 1 + 7.19T + 79T^{2} \) |
| 83 | \( 1 - 6.89T + 83T^{2} \) |
| 89 | \( 1 - 16.6T + 89T^{2} \) |
| 97 | \( 1 - 16.6T + 97T^{2} \) |
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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
−8.067913977679239379176583818489, −7.53964056076448151325785106541, −6.48235006332023444652276408936, −6.09759470077330154128721972006, −5.45308549600554707477218969232, −4.56820945742129502153473296111, −3.37361697944922703283881438169, −2.67672575298439177246183865916, −1.56684013257708014937215672977, −0.952063832262071991999442064159,
0.952063832262071991999442064159, 1.56684013257708014937215672977, 2.67672575298439177246183865916, 3.37361697944922703283881438169, 4.56820945742129502153473296111, 5.45308549600554707477218969232, 6.09759470077330154128721972006, 6.48235006332023444652276408936, 7.53964056076448151325785106541, 8.067913977679239379176583818489
