| L(s) = 1 | + 3·3-s + 5·5-s + 4.92·7-s + 9·9-s + 33.0·11-s + 13·13-s + 15·15-s − 119.·17-s − 127.·19-s + 14.7·21-s − 30.2·23-s + 25·25-s + 27·27-s − 309.·29-s − 57.9·31-s + 99.0·33-s + 24.6·35-s − 191.·37-s + 39·39-s + 78.4·41-s − 524.·43-s + 45·45-s + 183.·47-s − 318.·49-s − 358.·51-s − 241.·53-s + 165.·55-s + ⋯ |
| L(s) = 1 | + 0.577·3-s + 0.447·5-s + 0.265·7-s + 0.333·9-s + 0.905·11-s + 0.277·13-s + 0.258·15-s − 1.70·17-s − 1.54·19-s + 0.153·21-s − 0.274·23-s + 0.200·25-s + 0.192·27-s − 1.98·29-s − 0.335·31-s + 0.522·33-s + 0.118·35-s − 0.852·37-s + 0.160·39-s + 0.298·41-s − 1.86·43-s + 0.149·45-s + 0.568·47-s − 0.929·49-s − 0.983·51-s − 0.624·53-s + 0.404·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1560 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1560 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 3 | \( 1 - 3T \) |
| 5 | \( 1 - 5T \) |
| 13 | \( 1 - 13T \) |
| good | 7 | \( 1 - 4.92T + 343T^{2} \) |
| 11 | \( 1 - 33.0T + 1.33e3T^{2} \) |
| 17 | \( 1 + 119.T + 4.91e3T^{2} \) |
| 19 | \( 1 + 127.T + 6.85e3T^{2} \) |
| 23 | \( 1 + 30.2T + 1.21e4T^{2} \) |
| 29 | \( 1 + 309.T + 2.43e4T^{2} \) |
| 31 | \( 1 + 57.9T + 2.97e4T^{2} \) |
| 37 | \( 1 + 191.T + 5.06e4T^{2} \) |
| 41 | \( 1 - 78.4T + 6.89e4T^{2} \) |
| 43 | \( 1 + 524.T + 7.95e4T^{2} \) |
| 47 | \( 1 - 183.T + 1.03e5T^{2} \) |
| 53 | \( 1 + 241.T + 1.48e5T^{2} \) |
| 59 | \( 1 - 103.T + 2.05e5T^{2} \) |
| 61 | \( 1 - 778.T + 2.26e5T^{2} \) |
| 67 | \( 1 - 230.T + 3.00e5T^{2} \) |
| 71 | \( 1 - 668.T + 3.57e5T^{2} \) |
| 73 | \( 1 - 706.T + 3.89e5T^{2} \) |
| 79 | \( 1 + 283.T + 4.93e5T^{2} \) |
| 83 | \( 1 + 564.T + 5.71e5T^{2} \) |
| 89 | \( 1 + 1.56e3T + 7.04e5T^{2} \) |
| 97 | \( 1 - 1.44e3T + 9.12e5T^{2} \) |
| show more | |
| show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−8.766276261592593068394493947714, −8.083672215372119219727728523579, −6.85914603911816816107030070940, −6.48837168814449081755851458182, −5.34161911440084286926372407755, −4.28333898440720103814747400703, −3.64245304620427438416073953112, −2.21987922325415503628728471179, −1.68779430789988648162840868370, 0,
1.68779430789988648162840868370, 2.21987922325415503628728471179, 3.64245304620427438416073953112, 4.28333898440720103814747400703, 5.34161911440084286926372407755, 6.48837168814449081755851458182, 6.85914603911816816107030070940, 8.083672215372119219727728523579, 8.766276261592593068394493947714
