| L(s) = 1 | − 1.53·2-s − 2.66·3-s + 0.353·4-s + 4.08·6-s + 7-s + 2.52·8-s + 4.07·9-s − 5.84·11-s − 0.940·12-s − 0.0444·13-s − 1.53·14-s − 4.58·16-s + 7.20·17-s − 6.25·18-s − 2.29·19-s − 2.66·21-s + 8.97·22-s − 23-s − 6.71·24-s + 0.0681·26-s − 2.86·27-s + 0.353·28-s + 7.92·29-s − 6.44·31-s + 1.97·32-s + 15.5·33-s − 11.0·34-s + ⋯ |
| L(s) = 1 | − 1.08·2-s − 1.53·3-s + 0.176·4-s + 1.66·6-s + 0.377·7-s + 0.893·8-s + 1.35·9-s − 1.76·11-s − 0.271·12-s − 0.0123·13-s − 0.410·14-s − 1.14·16-s + 1.74·17-s − 1.47·18-s − 0.526·19-s − 0.580·21-s + 1.91·22-s − 0.208·23-s − 1.37·24-s + 0.0133·26-s − 0.551·27-s + 0.0668·28-s + 1.47·29-s − 1.15·31-s + 0.349·32-s + 2.70·33-s − 1.89·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4025 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4025 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 7 | \( 1 - T \) |
| 23 | \( 1 + T \) |
| good | 2 | \( 1 + 1.53T + 2T^{2} \) |
| 3 | \( 1 + 2.66T + 3T^{2} \) |
| 11 | \( 1 + 5.84T + 11T^{2} \) |
| 13 | \( 1 + 0.0444T + 13T^{2} \) |
| 17 | \( 1 - 7.20T + 17T^{2} \) |
| 19 | \( 1 + 2.29T + 19T^{2} \) |
| 29 | \( 1 - 7.92T + 29T^{2} \) |
| 31 | \( 1 + 6.44T + 31T^{2} \) |
| 37 | \( 1 + 0.590T + 37T^{2} \) |
| 41 | \( 1 + 11.0T + 41T^{2} \) |
| 43 | \( 1 - 3.01T + 43T^{2} \) |
| 47 | \( 1 + 9.08T + 47T^{2} \) |
| 53 | \( 1 + 5.31T + 53T^{2} \) |
| 59 | \( 1 + 3.75T + 59T^{2} \) |
| 61 | \( 1 - 12.6T + 61T^{2} \) |
| 67 | \( 1 - 1.58T + 67T^{2} \) |
| 71 | \( 1 + 2.44T + 71T^{2} \) |
| 73 | \( 1 - 1.09T + 73T^{2} \) |
| 79 | \( 1 - 0.301T + 79T^{2} \) |
| 83 | \( 1 - 12.3T + 83T^{2} \) |
| 89 | \( 1 - 6.18T + 89T^{2} \) |
| 97 | \( 1 - 15.4T + 97T^{2} \) |
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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
−8.022112604707800325751550364139, −7.54743656347220903800639827963, −6.69866253348086993786159993910, −5.79410907180210819874229942481, −5.05583081420037748348608647395, −4.77909104052563604668830255777, −3.38593616922889339328171468904, −2.00380889665107381092142607727, −0.923839097932536947928901606703, 0,
0.923839097932536947928901606703, 2.00380889665107381092142607727, 3.38593616922889339328171468904, 4.77909104052563604668830255777, 5.05583081420037748348608647395, 5.79410907180210819874229942481, 6.69866253348086993786159993910, 7.54743656347220903800639827963, 8.022112604707800325751550364139
