| L(s) = 1 | + 3·3-s − 5-s + 6·9-s + 5·11-s + 3·13-s − 3·15-s − 6·19-s − 6·23-s + 25-s + 9·27-s + 9·29-s − 4·31-s + 15·33-s − 2·37-s + 9·39-s − 4·41-s + 10·43-s − 6·45-s + 47-s + 4·53-s − 5·55-s − 18·57-s + 8·59-s − 8·61-s − 3·65-s + 12·67-s − 18·69-s + ⋯ |
| L(s) = 1 | + 1.73·3-s − 0.447·5-s + 2·9-s + 1.50·11-s + 0.832·13-s − 0.774·15-s − 1.37·19-s − 1.25·23-s + 1/5·25-s + 1.73·27-s + 1.67·29-s − 0.718·31-s + 2.61·33-s − 0.328·37-s + 1.44·39-s − 0.624·41-s + 1.52·43-s − 0.894·45-s + 0.145·47-s + 0.549·53-s − 0.674·55-s − 2.38·57-s + 1.04·59-s − 1.02·61-s − 0.372·65-s + 1.46·67-s − 2.16·69-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 283220 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 283220 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 5 | \( 1 + T \) |
| 7 | \( 1 \) |
| 17 | \( 1 \) |
| good | 3 | \( 1 - p T + p T^{2} \) |
| 11 | \( 1 - 5 T + p T^{2} \) |
| 13 | \( 1 - 3 T + p T^{2} \) |
| 19 | \( 1 + 6 T + p T^{2} \) |
| 23 | \( 1 + 6 T + p T^{2} \) |
| 29 | \( 1 - 9 T + p T^{2} \) |
| 31 | \( 1 + 4 T + p T^{2} \) |
| 37 | \( 1 + 2 T + p T^{2} \) |
| 41 | \( 1 + 4 T + p T^{2} \) |
| 43 | \( 1 - 10 T + p T^{2} \) |
| 47 | \( 1 - T + p T^{2} \) |
| 53 | \( 1 - 4 T + p T^{2} \) |
| 59 | \( 1 - 8 T + p T^{2} \) |
| 61 | \( 1 + 8 T + p T^{2} \) |
| 67 | \( 1 - 12 T + p T^{2} \) |
| 71 | \( 1 + 8 T + p T^{2} \) |
| 73 | \( 1 - 2 T + p T^{2} \) |
| 79 | \( 1 + 13 T + p T^{2} \) |
| 83 | \( 1 - 4 T + p T^{2} \) |
| 89 | \( 1 + 4 T + p T^{2} \) |
| 97 | \( 1 + 13 T + p T^{2} \) |
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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
−13.03809811863039, −12.47111280165947, −12.25947531689256, −11.62630326981033, −11.16553847017631, −10.43401412742863, −10.27253596491092, −9.515176151419551, −9.189689663960820, −8.672253272071377, −8.421110033392562, −8.089183258356828, −7.462607278316598, −6.897945945367457, −6.546677151313390, −6.072824127840814, −5.339843560794318, −4.339376446492818, −4.199392494983495, −3.814719065829667, −3.345614972725218, −2.592178837820522, −2.243258103650673, −1.450096527464436, −1.142194941601805, 0,
1.142194941601805, 1.450096527464436, 2.243258103650673, 2.592178837820522, 3.345614972725218, 3.814719065829667, 4.199392494983495, 4.339376446492818, 5.339843560794318, 6.072824127840814, 6.546677151313390, 6.897945945367457, 7.462607278316598, 8.089183258356828, 8.421110033392562, 8.672253272071377, 9.189689663960820, 9.515176151419551, 10.27253596491092, 10.43401412742863, 11.16553847017631, 11.62630326981033, 12.25947531689256, 12.47111280165947, 13.03809811863039
