L(s) = 1 | + 3·3-s − 3.29·5-s + 25.8·7-s + 9·9-s − 8.83·11-s + 13·13-s − 9.87·15-s − 10.2·17-s − 119.·19-s + 77.6·21-s + 141.·23-s − 114.·25-s + 27·27-s − 170.·29-s − 226.·31-s − 26.5·33-s − 85.1·35-s + 225.·37-s + 39·39-s − 274.·41-s − 111.·43-s − 29.6·45-s − 156.·47-s + 326.·49-s − 30.7·51-s + 85.1·53-s + 29.0·55-s + ⋯ |
L(s) = 1 | + 0.577·3-s − 0.294·5-s + 1.39·7-s + 0.333·9-s − 0.242·11-s + 0.277·13-s − 0.169·15-s − 0.146·17-s − 1.44·19-s + 0.806·21-s + 1.28·23-s − 0.913·25-s + 0.192·27-s − 1.09·29-s − 1.31·31-s − 0.139·33-s − 0.411·35-s + 1.00·37-s + 0.160·39-s − 1.04·41-s − 0.394·43-s − 0.0981·45-s − 0.485·47-s + 0.951·49-s − 0.0844·51-s + 0.220·53-s + 0.0712·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2496 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2496 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - 3T \) |
| 13 | \( 1 - 13T \) |
good | 5 | \( 1 + 3.29T + 125T^{2} \) |
| 7 | \( 1 - 25.8T + 343T^{2} \) |
| 11 | \( 1 + 8.83T + 1.33e3T^{2} \) |
| 17 | \( 1 + 10.2T + 4.91e3T^{2} \) |
| 19 | \( 1 + 119.T + 6.85e3T^{2} \) |
| 23 | \( 1 - 141.T + 1.21e4T^{2} \) |
| 29 | \( 1 + 170.T + 2.43e4T^{2} \) |
| 31 | \( 1 + 226.T + 2.97e4T^{2} \) |
| 37 | \( 1 - 225.T + 5.06e4T^{2} \) |
| 41 | \( 1 + 274.T + 6.89e4T^{2} \) |
| 43 | \( 1 + 111.T + 7.95e4T^{2} \) |
| 47 | \( 1 + 156.T + 1.03e5T^{2} \) |
| 53 | \( 1 - 85.1T + 1.48e5T^{2} \) |
| 59 | \( 1 + 889.T + 2.05e5T^{2} \) |
| 61 | \( 1 + 463.T + 2.26e5T^{2} \) |
| 67 | \( 1 + 459.T + 3.00e5T^{2} \) |
| 71 | \( 1 - 560.T + 3.57e5T^{2} \) |
| 73 | \( 1 - 784.T + 3.89e5T^{2} \) |
| 79 | \( 1 + 241.T + 4.93e5T^{2} \) |
| 83 | \( 1 + 1.27e3T + 5.71e5T^{2} \) |
| 89 | \( 1 - 1.08e3T + 7.04e5T^{2} \) |
| 97 | \( 1 + 79.9T + 9.12e5T^{2} \) |
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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
−8.121617874753475687427009663906, −7.67266448524076802886933105197, −6.82475704958091864019907538786, −5.77717529354572969361287225591, −4.87192587471788543110771209093, −4.21918831308961056947452332425, −3.31123904195694607949534632809, −2.12979498373865814289494293407, −1.48880635594307386183114939383, 0,
1.48880635594307386183114939383, 2.12979498373865814289494293407, 3.31123904195694607949534632809, 4.21918831308961056947452332425, 4.87192587471788543110771209093, 5.77717529354572969361287225591, 6.82475704958091864019907538786, 7.67266448524076802886933105197, 8.121617874753475687427009663906