| L(s) = 1 | − 0.618·3-s + 1.61·5-s − 2.61·9-s − 5.23·11-s + 3.23·13-s − 1.00·15-s − 17-s − 0.472·19-s + 5.70·23-s − 2.38·25-s + 3.47·27-s + 7.70·29-s + 9.32·31-s + 3.23·33-s − 8.47·37-s − 2.00·39-s − 11.0·41-s − 0.909·43-s − 4.23·45-s − 0.472·47-s + 0.618·51-s − 13.7·53-s − 8.47·55-s + 0.291·57-s − 8.32·61-s + 5.23·65-s − 1.90·67-s + ⋯ |
| L(s) = 1 | − 0.356·3-s + 0.723·5-s − 0.872·9-s − 1.57·11-s + 0.897·13-s − 0.258·15-s − 0.242·17-s − 0.108·19-s + 1.19·23-s − 0.476·25-s + 0.668·27-s + 1.43·29-s + 1.67·31-s + 0.563·33-s − 1.39·37-s − 0.320·39-s − 1.73·41-s − 0.138·43-s − 0.631·45-s − 0.0688·47-s + 0.0865·51-s − 1.89·53-s − 1.14·55-s + 0.0386·57-s − 1.06·61-s + 0.649·65-s − 0.233·67-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3332 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3332 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 2 | \( 1 \) |
| 7 | \( 1 \) |
| 17 | \( 1 + T \) |
| good | 3 | \( 1 + 0.618T + 3T^{2} \) |
| 5 | \( 1 - 1.61T + 5T^{2} \) |
| 11 | \( 1 + 5.23T + 11T^{2} \) |
| 13 | \( 1 - 3.23T + 13T^{2} \) |
| 19 | \( 1 + 0.472T + 19T^{2} \) |
| 23 | \( 1 - 5.70T + 23T^{2} \) |
| 29 | \( 1 - 7.70T + 29T^{2} \) |
| 31 | \( 1 - 9.32T + 31T^{2} \) |
| 37 | \( 1 + 8.47T + 37T^{2} \) |
| 41 | \( 1 + 11.0T + 41T^{2} \) |
| 43 | \( 1 + 0.909T + 43T^{2} \) |
| 47 | \( 1 + 0.472T + 47T^{2} \) |
| 53 | \( 1 + 13.7T + 53T^{2} \) |
| 59 | \( 1 + 59T^{2} \) |
| 61 | \( 1 + 8.32T + 61T^{2} \) |
| 67 | \( 1 + 1.90T + 67T^{2} \) |
| 71 | \( 1 + 6.94T + 71T^{2} \) |
| 73 | \( 1 + 13.8T + 73T^{2} \) |
| 79 | \( 1 + 14.9T + 79T^{2} \) |
| 83 | \( 1 - 11.4T + 83T^{2} \) |
| 89 | \( 1 + 2T + 89T^{2} \) |
| 97 | \( 1 + 3.90T + 97T^{2} \) |
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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
−8.533732283680664436577677949803, −7.54781745966044508964304292064, −6.49150780981797374005224583921, −6.06228527587588333025655071844, −5.16606883853046922716908066146, −4.73395582670959533051629313518, −3.20069528201449284083376971016, −2.69119096772723637478398714994, −1.45218580027003231569425818843, 0,
1.45218580027003231569425818843, 2.69119096772723637478398714994, 3.20069528201449284083376971016, 4.73395582670959533051629313518, 5.16606883853046922716908066146, 6.06228527587588333025655071844, 6.49150780981797374005224583921, 7.54781745966044508964304292064, 8.533732283680664436577677949803
