| 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 + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 312 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 312 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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
−10.32562671646902159028816148497, −9.993498419882439564928277611599, −8.911621503580280151647705856309, −8.028751454148506341239027958314, −6.73373998475477755736772247052, −6.05350067582724653057719832987, −4.45692626904635966156228758992, −3.26800874539872326447758293689, −2.13155032319669596782182198073, 0,
2.13155032319669596782182198073, 3.26800874539872326447758293689, 4.45692626904635966156228758992, 6.05350067582724653057719832987, 6.73373998475477755736772247052, 8.028751454148506341239027958314, 8.911621503580280151647705856309, 9.993498419882439564928277611599, 10.32562671646902159028816148497
