| L(s) = 1 | + 3-s − 2·5-s + 9-s + 13-s − 2·15-s + 17-s − 25-s + 27-s − 2·29-s − 8·31-s + 10·37-s + 39-s − 6·41-s + 4·43-s − 2·45-s + 8·47-s − 7·49-s + 51-s − 2·53-s + 4·59-s − 10·61-s − 2·65-s − 8·67-s + 12·71-s + 6·73-s − 75-s + 8·79-s + ⋯ |
| L(s) = 1 | + 0.577·3-s − 0.894·5-s + 1/3·9-s + 0.277·13-s − 0.516·15-s + 0.242·17-s − 1/5·25-s + 0.192·27-s − 0.371·29-s − 1.43·31-s + 1.64·37-s + 0.160·39-s − 0.937·41-s + 0.609·43-s − 0.298·45-s + 1.16·47-s − 49-s + 0.140·51-s − 0.274·53-s + 0.520·59-s − 1.28·61-s − 0.248·65-s − 0.977·67-s + 1.42·71-s + 0.702·73-s − 0.115·75-s + 0.900·79-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 10608 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 10608 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 3 | \( 1 - T \) |
| 13 | \( 1 - T \) |
| 17 | \( 1 - T \) |
| good | 5 | \( 1 + 2 T + p T^{2} \) |
| 7 | \( 1 + p T^{2} \) |
| 11 | \( 1 + p T^{2} \) |
| 19 | \( 1 + p T^{2} \) |
| 23 | \( 1 + p T^{2} \) |
| 29 | \( 1 + 2 T + p T^{2} \) |
| 31 | \( 1 + 8 T + p T^{2} \) |
| 37 | \( 1 - 10 T + p T^{2} \) |
| 41 | \( 1 + 6 T + p T^{2} \) |
| 43 | \( 1 - 4 T + p T^{2} \) |
| 47 | \( 1 - 8 T + p T^{2} \) |
| 53 | \( 1 + 2 T + p T^{2} \) |
| 59 | \( 1 - 4 T + p T^{2} \) |
| 61 | \( 1 + 10 T + p T^{2} \) |
| 67 | \( 1 + 8 T + p T^{2} \) |
| 71 | \( 1 - 12 T + p T^{2} \) |
| 73 | \( 1 - 6 T + p T^{2} \) |
| 79 | \( 1 - 8 T + p T^{2} \) |
| 83 | \( 1 + 12 T + p T^{2} \) |
| 89 | \( 1 + 10 T + p T^{2} \) |
| 97 | \( 1 - 6 T + p T^{2} \) |
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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
−16.66631276301960, −16.25680787915157, −15.57714539682043, −15.17173643374878, −14.62862642683339, −14.04311309691119, −13.42117435676092, −12.78570194725759, −12.31927875172950, −11.57477684987254, −11.09536050532706, −10.50253105349281, −9.629331687041868, −9.216103931125854, −8.479348659581446, −7.831906000940490, −7.517740198431699, −6.728572761812273, −5.937263950072954, −5.180027146916564, −4.270017144904502, −3.797251210256854, −3.103426459382065, −2.216716557049049, −1.230491871107509, 0,
1.230491871107509, 2.216716557049049, 3.103426459382065, 3.797251210256854, 4.270017144904502, 5.180027146916564, 5.937263950072954, 6.728572761812273, 7.517740198431699, 7.831906000940490, 8.479348659581446, 9.216103931125854, 9.629331687041868, 10.50253105349281, 11.09536050532706, 11.57477684987254, 12.31927875172950, 12.78570194725759, 13.42117435676092, 14.04311309691119, 14.62862642683339, 15.17173643374878, 15.57714539682043, 16.25680787915157, 16.66631276301960
