| L(s) = 1 | − 2·2-s + 0.843·3-s + 4·4-s + 13.2·5-s − 1.68·6-s − 8·8-s − 26.2·9-s − 26.4·10-s − 2.18·11-s + 3.37·12-s + 53.8·13-s + 11.1·15-s + 16·16-s − 15.9·17-s + 52.5·18-s − 110.·19-s + 52.9·20-s + 4.36·22-s + 23·23-s − 6.74·24-s + 49.9·25-s − 107.·26-s − 44.9·27-s − 192.·29-s − 22.3·30-s + 177.·31-s − 32·32-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 0.162·3-s + 0.5·4-s + 1.18·5-s − 0.114·6-s − 0.353·8-s − 0.973·9-s − 0.836·10-s − 0.0598·11-s + 0.0811·12-s + 1.14·13-s + 0.192·15-s + 0.250·16-s − 0.228·17-s + 0.688·18-s − 1.33·19-s + 0.591·20-s + 0.0423·22-s + 0.208·23-s − 0.0574·24-s + 0.399·25-s − 0.812·26-s − 0.320·27-s − 1.23·29-s − 0.135·30-s + 1.03·31-s − 0.176·32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2254 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2254 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + 2T \) |
| 7 | \( 1 \) |
| 23 | \( 1 - 23T \) |
| good | 3 | \( 1 - 0.843T + 27T^{2} \) |
| 5 | \( 1 - 13.2T + 125T^{2} \) |
| 11 | \( 1 + 2.18T + 1.33e3T^{2} \) |
| 13 | \( 1 - 53.8T + 2.19e3T^{2} \) |
| 17 | \( 1 + 15.9T + 4.91e3T^{2} \) |
| 19 | \( 1 + 110.T + 6.85e3T^{2} \) |
| 29 | \( 1 + 192.T + 2.43e4T^{2} \) |
| 31 | \( 1 - 177.T + 2.97e4T^{2} \) |
| 37 | \( 1 - 418.T + 5.06e4T^{2} \) |
| 41 | \( 1 + 134.T + 6.89e4T^{2} \) |
| 43 | \( 1 - 304.T + 7.95e4T^{2} \) |
| 47 | \( 1 + 399.T + 1.03e5T^{2} \) |
| 53 | \( 1 + 385.T + 1.48e5T^{2} \) |
| 59 | \( 1 + 284.T + 2.05e5T^{2} \) |
| 61 | \( 1 + 75.7T + 2.26e5T^{2} \) |
| 67 | \( 1 + 457.T + 3.00e5T^{2} \) |
| 71 | \( 1 - 563.T + 3.57e5T^{2} \) |
| 73 | \( 1 + 366.T + 3.89e5T^{2} \) |
| 79 | \( 1 + 104.T + 4.93e5T^{2} \) |
| 83 | \( 1 - 691.T + 5.71e5T^{2} \) |
| 89 | \( 1 + 898.T + 7.04e5T^{2} \) |
| 97 | \( 1 + 276.T + 9.12e5T^{2} \) |
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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
−8.378714214847883487262508466492, −7.81358115058318626696529906353, −6.43455999841413707046175616263, −6.21496879724959795753427558262, −5.40209972682862648503996600297, −4.16983524196346113599963775150, −2.98049538951720334060950794397, −2.20671517104535663693327915149, −1.30960795870655493239074371119, 0,
1.30960795870655493239074371119, 2.20671517104535663693327915149, 2.98049538951720334060950794397, 4.16983524196346113599963775150, 5.40209972682862648503996600297, 6.21496879724959795753427558262, 6.43455999841413707046175616263, 7.81358115058318626696529906353, 8.378714214847883487262508466492
