| L(s) = 1 | + 0.783·2-s + 0.544·3-s − 1.38·4-s + 0.426·6-s + 1.18·7-s − 2.65·8-s − 2.70·9-s + 2.55·11-s − 0.755·12-s − 0.368·13-s + 0.931·14-s + 0.693·16-s − 2.11·18-s + 6.61·19-s + 0.647·21-s + 2.00·22-s − 3.86·23-s − 1.44·24-s − 0.288·26-s − 3.10·27-s − 1.64·28-s − 2.31·29-s − 6.62·31-s + 5.84·32-s + 1.39·33-s + 3.74·36-s + 3.17·37-s + ⋯ |
| L(s) = 1 | + 0.554·2-s + 0.314·3-s − 0.693·4-s + 0.174·6-s + 0.449·7-s − 0.937·8-s − 0.901·9-s + 0.770·11-s − 0.218·12-s − 0.102·13-s + 0.248·14-s + 0.173·16-s − 0.499·18-s + 1.51·19-s + 0.141·21-s + 0.426·22-s − 0.805·23-s − 0.295·24-s − 0.0565·26-s − 0.598·27-s − 0.311·28-s − 0.429·29-s − 1.18·31-s + 1.03·32-s + 0.242·33-s + 0.624·36-s + 0.522·37-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
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 - 0.783T + 2T^{2} \) |
| 3 | \( 1 - 0.544T + 3T^{2} \) |
| 7 | \( 1 - 1.18T + 7T^{2} \) |
| 11 | \( 1 - 2.55T + 11T^{2} \) |
| 13 | \( 1 + 0.368T + 13T^{2} \) |
| 19 | \( 1 - 6.61T + 19T^{2} \) |
| 23 | \( 1 + 3.86T + 23T^{2} \) |
| 29 | \( 1 + 2.31T + 29T^{2} \) |
| 31 | \( 1 + 6.62T + 31T^{2} \) |
| 37 | \( 1 - 3.17T + 37T^{2} \) |
| 41 | \( 1 + 7.30T + 41T^{2} \) |
| 43 | \( 1 + 6.82T + 43T^{2} \) |
| 47 | \( 1 - 7.80T + 47T^{2} \) |
| 53 | \( 1 + 8.01T + 53T^{2} \) |
| 59 | \( 1 - 5.22T + 59T^{2} \) |
| 61 | \( 1 - 8.12T + 61T^{2} \) |
| 67 | \( 1 - 7.94T + 67T^{2} \) |
| 71 | \( 1 + 11.8T + 71T^{2} \) |
| 73 | \( 1 - 14.7T + 73T^{2} \) |
| 79 | \( 1 + 0.813T + 79T^{2} \) |
| 83 | \( 1 + 3.99T + 83T^{2} \) |
| 89 | \( 1 + 9.14T + 89T^{2} \) |
| 97 | \( 1 - 7.06T + 97T^{2} \) |
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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
−7.70255253867669179709070327194, −6.81608882605194404409014495556, −5.93078112780240488832002601394, −5.38126008286278798397798607371, −4.81172112911462918909607371975, −3.75566015849714826900067001930, −3.46793600608614257647097911906, −2.43953792000200586841426749698, −1.32309402360652753363243493078, 0,
1.32309402360652753363243493078, 2.43953792000200586841426749698, 3.46793600608614257647097911906, 3.75566015849714826900067001930, 4.81172112911462918909607371975, 5.38126008286278798397798607371, 5.93078112780240488832002601394, 6.81608882605194404409014495556, 7.70255253867669179709070327194
