| L(s) = 1 | − 1.43·2-s + 1.29·3-s + 0.0450·4-s + 5-s − 1.85·6-s + 2.79·8-s − 1.32·9-s − 1.43·10-s − 0.314·11-s + 0.0583·12-s − 4.54·13-s + 1.29·15-s − 4.08·16-s + 6.35·17-s + 1.89·18-s − 4.10·19-s + 0.0450·20-s + 0.449·22-s + 23-s + 3.61·24-s + 25-s + 6.49·26-s − 5.59·27-s − 0.908·29-s − 1.85·30-s + 5.45·31-s + 0.254·32-s + ⋯ |
| L(s) = 1 | − 1.01·2-s + 0.747·3-s + 0.0225·4-s + 0.447·5-s − 0.755·6-s + 0.988·8-s − 0.441·9-s − 0.452·10-s − 0.0948·11-s + 0.0168·12-s − 1.26·13-s + 0.334·15-s − 1.02·16-s + 1.54·17-s + 0.446·18-s − 0.942·19-s + 0.0100·20-s + 0.0959·22-s + 0.208·23-s + 0.738·24-s + 0.200·25-s + 1.27·26-s − 1.07·27-s − 0.168·29-s − 0.337·30-s + 0.979·31-s + 0.0450·32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5635 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5635 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) |
| 7 | \( 1 \) |
| 23 | \( 1 - T \) |
| good | 2 | \( 1 + 1.43T + 2T^{2} \) |
| 3 | \( 1 - 1.29T + 3T^{2} \) |
| 11 | \( 1 + 0.314T + 11T^{2} \) |
| 13 | \( 1 + 4.54T + 13T^{2} \) |
| 17 | \( 1 - 6.35T + 17T^{2} \) |
| 19 | \( 1 + 4.10T + 19T^{2} \) |
| 29 | \( 1 + 0.908T + 29T^{2} \) |
| 31 | \( 1 - 5.45T + 31T^{2} \) |
| 37 | \( 1 - 1.06T + 37T^{2} \) |
| 41 | \( 1 - 5.54T + 41T^{2} \) |
| 43 | \( 1 + 6.78T + 43T^{2} \) |
| 47 | \( 1 - 2.37T + 47T^{2} \) |
| 53 | \( 1 - 1.63T + 53T^{2} \) |
| 59 | \( 1 - 3.47T + 59T^{2} \) |
| 61 | \( 1 + 12.9T + 61T^{2} \) |
| 67 | \( 1 - 1.19T + 67T^{2} \) |
| 71 | \( 1 - 9.70T + 71T^{2} \) |
| 73 | \( 1 + 15.9T + 73T^{2} \) |
| 79 | \( 1 + 2.26T + 79T^{2} \) |
| 83 | \( 1 + 2.19T + 83T^{2} \) |
| 89 | \( 1 - 13.8T + 89T^{2} \) |
| 97 | \( 1 - 2.77T + 97T^{2} \) |
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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
−7.903367973281372143075180002542, −7.48906111170740532154871594750, −6.55689323161986741876248118682, −5.60244459604218104775372115222, −4.90671417347122846523997556675, −4.01855739536368126741291430878, −2.94809364141824016923682188558, −2.29273476669142699476123667620, −1.27220400658272979594852797464, 0,
1.27220400658272979594852797464, 2.29273476669142699476123667620, 2.94809364141824016923682188558, 4.01855739536368126741291430878, 4.90671417347122846523997556675, 5.60244459604218104775372115222, 6.55689323161986741876248118682, 7.48906111170740532154871594750, 7.903367973281372143075180002542
