| L(s) = 1 | + 2·2-s − 3·3-s + 4·4-s + 5·5-s − 6·6-s + 8·8-s + 9·9-s + 10·10-s + 56·11-s − 12·12-s − 54·13-s − 15·15-s + 16·16-s − 94·17-s + 18·18-s − 36·19-s + 20·20-s + 112·22-s − 84·23-s − 24·24-s + 25·25-s − 108·26-s − 27·27-s − 258·29-s − 30·30-s + 40·31-s + 32·32-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 0.577·3-s + 1/2·4-s + 0.447·5-s − 0.408·6-s + 0.353·8-s + 1/3·9-s + 0.316·10-s + 1.53·11-s − 0.288·12-s − 1.15·13-s − 0.258·15-s + 1/4·16-s − 1.34·17-s + 0.235·18-s − 0.434·19-s + 0.223·20-s + 1.08·22-s − 0.761·23-s − 0.204·24-s + 1/5·25-s − 0.814·26-s − 0.192·27-s − 1.65·29-s − 0.182·30-s + 0.231·31-s + 0.176·32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1470 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1470 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - p T \) |
| 3 | \( 1 + p T \) |
| 5 | \( 1 - p T \) |
| 7 | \( 1 \) |
| good | 11 | \( 1 - 56 T + p^{3} T^{2} \) |
| 13 | \( 1 + 54 T + p^{3} T^{2} \) |
| 17 | \( 1 + 94 T + p^{3} T^{2} \) |
| 19 | \( 1 + 36 T + p^{3} T^{2} \) |
| 23 | \( 1 + 84 T + p^{3} T^{2} \) |
| 29 | \( 1 + 258 T + p^{3} T^{2} \) |
| 31 | \( 1 - 40 T + p^{3} T^{2} \) |
| 37 | \( 1 + 178 T + p^{3} T^{2} \) |
| 41 | \( 1 - 146 T + p^{3} T^{2} \) |
| 43 | \( 1 - 148 T + p^{3} T^{2} \) |
| 47 | \( 1 - 200 T + p^{3} T^{2} \) |
| 53 | \( 1 + 130 T + p^{3} T^{2} \) |
| 59 | \( 1 + 188 T + p^{3} T^{2} \) |
| 61 | \( 1 + 94 T + p^{3} T^{2} \) |
| 67 | \( 1 + 444 T + p^{3} T^{2} \) |
| 71 | \( 1 - 532 T + p^{3} T^{2} \) |
| 73 | \( 1 + 770 T + p^{3} T^{2} \) |
| 79 | \( 1 + 536 T + p^{3} T^{2} \) |
| 83 | \( 1 - 1076 T + p^{3} T^{2} \) |
| 89 | \( 1 - 1090 T + p^{3} T^{2} \) |
| 97 | \( 1 + 1274 T + p^{3} T^{2} \) |
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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
−8.983363977445803728864987550392, −7.64269446717650640632318950509, −6.81775288831626796399210017556, −6.26414289182205909739775392937, −5.43221748796815043662894208216, −4.46966882022820589124299181946, −3.86843092728511390107728870810, −2.42146267551731254518732782757, −1.56802285955720430300042401170, 0,
1.56802285955720430300042401170, 2.42146267551731254518732782757, 3.86843092728511390107728870810, 4.46966882022820589124299181946, 5.43221748796815043662894208216, 6.26414289182205909739775392937, 6.81775288831626796399210017556, 7.64269446717650640632318950509, 8.983363977445803728864987550392
