| L(s) = 1 | − 2.55·2-s + 2.08·3-s + 4.54·4-s − 5.33·6-s + 0.752·7-s − 6.51·8-s + 1.34·9-s − 6.22·11-s + 9.47·12-s − 6.48·13-s − 1.92·14-s + 7.56·16-s − 3.43·18-s − 6.05·19-s + 1.56·21-s + 15.9·22-s − 2.69·23-s − 13.5·24-s + 16.5·26-s − 3.45·27-s + 3.42·28-s + 5.99·29-s − 3.55·31-s − 6.33·32-s − 12.9·33-s + 6.10·36-s − 0.0501·37-s + ⋯ |
| L(s) = 1 | − 1.80·2-s + 1.20·3-s + 2.27·4-s − 2.17·6-s + 0.284·7-s − 2.30·8-s + 0.448·9-s − 1.87·11-s + 2.73·12-s − 1.79·13-s − 0.514·14-s + 1.89·16-s − 0.810·18-s − 1.38·19-s + 0.342·21-s + 3.39·22-s − 0.562·23-s − 2.77·24-s + 3.25·26-s − 0.664·27-s + 0.646·28-s + 1.11·29-s − 0.639·31-s − 1.12·32-s − 2.25·33-s + 1.01·36-s − 0.00823·37-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 7225 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7225 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.5960830377\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.5960830377\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 \) |
| 17 | \( 1 \) |
| good | 2 | \( 1 + 2.55T + 2T^{2} \) |
| 3 | \( 1 - 2.08T + 3T^{2} \) |
| 7 | \( 1 - 0.752T + 7T^{2} \) |
| 11 | \( 1 + 6.22T + 11T^{2} \) |
| 13 | \( 1 + 6.48T + 13T^{2} \) |
| 19 | \( 1 + 6.05T + 19T^{2} \) |
| 23 | \( 1 + 2.69T + 23T^{2} \) |
| 29 | \( 1 - 5.99T + 29T^{2} \) |
| 31 | \( 1 + 3.55T + 31T^{2} \) |
| 37 | \( 1 + 0.0501T + 37T^{2} \) |
| 41 | \( 1 - 5.54T + 41T^{2} \) |
| 43 | \( 1 - 8.27T + 43T^{2} \) |
| 47 | \( 1 - 7.32T + 47T^{2} \) |
| 53 | \( 1 - 5.75T + 53T^{2} \) |
| 59 | \( 1 - 2.54T + 59T^{2} \) |
| 61 | \( 1 - 1.87T + 61T^{2} \) |
| 67 | \( 1 - 2.41T + 67T^{2} \) |
| 71 | \( 1 + 6.73T + 71T^{2} \) |
| 73 | \( 1 - 12.0T + 73T^{2} \) |
| 79 | \( 1 + 4.70T + 79T^{2} \) |
| 83 | \( 1 + 0.958T + 83T^{2} \) |
| 89 | \( 1 + 0.208T + 89T^{2} \) |
| 97 | \( 1 + 5.39T + 97T^{2} \) |
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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
−8.063108538399138496279931109328, −7.54506632022784151104892869385, −7.15975312222850150099231038231, −6.08689238832541600504510681125, −5.17954541045738119210193549971, −4.22551801617529858296202712472, −2.85738022036651799864966272761, −2.43066146793304381187864886878, −2.02946588837163862766162063956, −0.44191393354555008073611496356,
0.44191393354555008073611496356, 2.02946588837163862766162063956, 2.43066146793304381187864886878, 2.85738022036651799864966272761, 4.22551801617529858296202712472, 5.17954541045738119210193549971, 6.08689238832541600504510681125, 7.15975312222850150099231038231, 7.54506632022784151104892869385, 8.063108538399138496279931109328
