| L(s) = 1 | − 2.37·2-s + 1.76·3-s + 3.61·4-s + 2.72·5-s − 4.17·6-s + 3.47·7-s − 3.83·8-s + 0.103·9-s − 6.45·10-s + 1.70·11-s + 6.37·12-s + 5.48·13-s − 8.23·14-s + 4.80·15-s + 1.85·16-s + 2.33·17-s − 0.245·18-s − 5.33·19-s + 9.85·20-s + 6.11·21-s − 4.03·22-s + 4.15·23-s − 6.75·24-s + 2.42·25-s − 12.9·26-s − 5.10·27-s + 12.5·28-s + ⋯ |
| L(s) = 1 | − 1.67·2-s + 1.01·3-s + 1.80·4-s + 1.21·5-s − 1.70·6-s + 1.31·7-s − 1.35·8-s + 0.0345·9-s − 2.04·10-s + 0.513·11-s + 1.83·12-s + 1.52·13-s − 2.20·14-s + 1.23·15-s + 0.462·16-s + 0.565·17-s − 0.0579·18-s − 1.22·19-s + 2.20·20-s + 1.33·21-s − 0.860·22-s + 0.867·23-s − 1.37·24-s + 0.485·25-s − 2.54·26-s − 0.981·27-s + 2.37·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1849 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1849 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.783692608\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.783692608\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 43 | \( 1 \) |
| good | 2 | \( 1 + 2.37T + 2T^{2} \) |
| 3 | \( 1 - 1.76T + 3T^{2} \) |
| 5 | \( 1 - 2.72T + 5T^{2} \) |
| 7 | \( 1 - 3.47T + 7T^{2} \) |
| 11 | \( 1 - 1.70T + 11T^{2} \) |
| 13 | \( 1 - 5.48T + 13T^{2} \) |
| 17 | \( 1 - 2.33T + 17T^{2} \) |
| 19 | \( 1 + 5.33T + 19T^{2} \) |
| 23 | \( 1 - 4.15T + 23T^{2} \) |
| 29 | \( 1 - 0.689T + 29T^{2} \) |
| 31 | \( 1 + 0.844T + 31T^{2} \) |
| 37 | \( 1 + 2.19T + 37T^{2} \) |
| 41 | \( 1 + 2.05T + 41T^{2} \) |
| 47 | \( 1 - 3.81T + 47T^{2} \) |
| 53 | \( 1 - 7.62T + 53T^{2} \) |
| 59 | \( 1 - 4.47T + 59T^{2} \) |
| 61 | \( 1 + 14.7T + 61T^{2} \) |
| 67 | \( 1 - 9.78T + 67T^{2} \) |
| 71 | \( 1 + 4.87T + 71T^{2} \) |
| 73 | \( 1 - 2.56T + 73T^{2} \) |
| 79 | \( 1 + 10.7T + 79T^{2} \) |
| 83 | \( 1 + 10.1T + 83T^{2} \) |
| 89 | \( 1 + 11.4T + 89T^{2} \) |
| 97 | \( 1 + 9.71T + 97T^{2} \) |
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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
−9.004017528360311274103518852513, −8.578752872196237650193847297033, −8.173379510881022929826115293920, −7.21420229353915427842255557954, −6.30931794767761620216696888934, −5.48749830987645066301514980829, −4.08382905864648710627446057880, −2.76583595097158337211489595154, −1.81899723996019667042294949756, −1.30606760638705983726352778001,
1.30606760638705983726352778001, 1.81899723996019667042294949756, 2.76583595097158337211489595154, 4.08382905864648710627446057880, 5.48749830987645066301514980829, 6.30931794767761620216696888934, 7.21420229353915427842255557954, 8.173379510881022929826115293920, 8.578752872196237650193847297033, 9.004017528360311274103518852513
