| L(s) = 1 | − 2·2-s − 3·3-s + 4·4-s + 14.6·5-s + 6·6-s + 12.1·7-s − 8·8-s + 9·9-s − 29.2·10-s − 19.6·11-s − 12·12-s + 9.08·13-s − 24.3·14-s − 43.8·15-s + 16·16-s − 18·18-s − 56.4·19-s + 58.5·20-s − 36.4·21-s + 39.2·22-s + 119.·23-s + 24·24-s + 89.1·25-s − 18.1·26-s − 27·27-s + 48.6·28-s − 98.5·29-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 0.577·3-s + 0.5·4-s + 1.30·5-s + 0.408·6-s + 0.656·7-s − 0.353·8-s + 0.333·9-s − 0.925·10-s − 0.537·11-s − 0.288·12-s + 0.193·13-s − 0.464·14-s − 0.755·15-s + 0.250·16-s − 0.235·18-s − 0.681·19-s + 0.654·20-s − 0.379·21-s + 0.380·22-s + 1.08·23-s + 0.204·24-s + 0.713·25-s − 0.137·26-s − 0.192·27-s + 0.328·28-s − 0.631·29-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1734 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1734 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + 2T \) |
| 3 | \( 1 + 3T \) |
| 17 | \( 1 \) |
| good | 5 | \( 1 - 14.6T + 125T^{2} \) |
| 7 | \( 1 - 12.1T + 343T^{2} \) |
| 11 | \( 1 + 19.6T + 1.33e3T^{2} \) |
| 13 | \( 1 - 9.08T + 2.19e3T^{2} \) |
| 19 | \( 1 + 56.4T + 6.85e3T^{2} \) |
| 23 | \( 1 - 119.T + 1.21e4T^{2} \) |
| 29 | \( 1 + 98.5T + 2.43e4T^{2} \) |
| 31 | \( 1 - 48.3T + 2.97e4T^{2} \) |
| 37 | \( 1 - 1.86T + 5.06e4T^{2} \) |
| 41 | \( 1 + 385.T + 6.89e4T^{2} \) |
| 43 | \( 1 + 420.T + 7.95e4T^{2} \) |
| 47 | \( 1 + 300.T + 1.03e5T^{2} \) |
| 53 | \( 1 + 491.T + 1.48e5T^{2} \) |
| 59 | \( 1 - 450.T + 2.05e5T^{2} \) |
| 61 | \( 1 + 683.T + 2.26e5T^{2} \) |
| 67 | \( 1 + 394.T + 3.00e5T^{2} \) |
| 71 | \( 1 + 73.1T + 3.57e5T^{2} \) |
| 73 | \( 1 - 706.T + 3.89e5T^{2} \) |
| 79 | \( 1 - 587.T + 4.93e5T^{2} \) |
| 83 | \( 1 + 1.32e3T + 5.71e5T^{2} \) |
| 89 | \( 1 - 1.10e3T + 7.04e5T^{2} \) |
| 97 | \( 1 + 312.T + 9.12e5T^{2} \) |
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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
−8.609124903893718058697980246602, −7.893369084358209257053816870363, −6.83394333028547051436333933346, −6.26407355415293424689878621402, −5.34679140480882080573843806708, −4.77677624681017848047569975927, −3.20261306253635515138626722000, −2.00890389901125882025926916243, −1.39165642838734712795446036768, 0,
1.39165642838734712795446036768, 2.00890389901125882025926916243, 3.20261306253635515138626722000, 4.77677624681017848047569975927, 5.34679140480882080573843806708, 6.26407355415293424689878621402, 6.83394333028547051436333933346, 7.893369084358209257053816870363, 8.609124903893718058697980246602
