| L(s) = 1 | + 3·3-s + 5·5-s + 4·7-s + 9·9-s − 72·11-s + 6·13-s + 15·15-s + 38·17-s − 52·19-s + 12·21-s + 152·23-s + 25·25-s + 27·27-s + 78·29-s + 120·31-s − 216·33-s + 20·35-s + 150·37-s + 18·39-s + 362·41-s + 484·43-s + 45·45-s + 280·47-s − 327·49-s + 114·51-s + 670·53-s − 360·55-s + ⋯ |
| L(s) = 1 | + 0.577·3-s + 0.447·5-s + 0.215·7-s + 1/3·9-s − 1.97·11-s + 0.128·13-s + 0.258·15-s + 0.542·17-s − 0.627·19-s + 0.124·21-s + 1.37·23-s + 1/5·25-s + 0.192·27-s + 0.499·29-s + 0.695·31-s − 1.13·33-s + 0.0965·35-s + 0.666·37-s + 0.0739·39-s + 1.37·41-s + 1.71·43-s + 0.149·45-s + 0.868·47-s − 0.953·49-s + 0.313·51-s + 1.73·53-s − 0.882·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 960 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 960 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(2.721479868\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.721479868\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 - p T \) |
| 5 | \( 1 - p T \) |
| good | 7 | \( 1 - 4 T + p^{3} T^{2} \) |
| 11 | \( 1 + 72 T + p^{3} T^{2} \) |
| 13 | \( 1 - 6 T + p^{3} T^{2} \) |
| 17 | \( 1 - 38 T + p^{3} T^{2} \) |
| 19 | \( 1 + 52 T + p^{3} T^{2} \) |
| 23 | \( 1 - 152 T + p^{3} T^{2} \) |
| 29 | \( 1 - 78 T + p^{3} T^{2} \) |
| 31 | \( 1 - 120 T + p^{3} T^{2} \) |
| 37 | \( 1 - 150 T + p^{3} T^{2} \) |
| 41 | \( 1 - 362 T + p^{3} T^{2} \) |
| 43 | \( 1 - 484 T + p^{3} T^{2} \) |
| 47 | \( 1 - 280 T + p^{3} T^{2} \) |
| 53 | \( 1 - 670 T + p^{3} T^{2} \) |
| 59 | \( 1 + 696 T + p^{3} T^{2} \) |
| 61 | \( 1 + 222 T + p^{3} T^{2} \) |
| 67 | \( 1 - 4 T + p^{3} T^{2} \) |
| 71 | \( 1 - 96 T + p^{3} T^{2} \) |
| 73 | \( 1 - 178 T + p^{3} T^{2} \) |
| 79 | \( 1 + 8 p T + p^{3} T^{2} \) |
| 83 | \( 1 - 612 T + p^{3} T^{2} \) |
| 89 | \( 1 - 994 T + p^{3} T^{2} \) |
| 97 | \( 1 - 1634 T + p^{3} T^{2} \) |
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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
−9.602833253589732677038361835420, −8.797713039137442986696093513351, −7.927174504410886966377397851164, −7.37583096620943058861132807385, −6.13606578440774573974301392929, −5.25593762608916136706930174278, −4.39105529606119892485616907639, −2.94823415165923597331657264831, −2.37069698248616616567054217194, −0.868365486036512624391303602934,
0.868365486036512624391303602934, 2.37069698248616616567054217194, 2.94823415165923597331657264831, 4.39105529606119892485616907639, 5.25593762608916136706930174278, 6.13606578440774573974301392929, 7.37583096620943058861132807385, 7.927174504410886966377397851164, 8.797713039137442986696093513351, 9.602833253589732677038361835420
