| L(s) = 1 | − 3-s − 2.70·5-s − 7-s + 9-s + 4·11-s − 0.701·13-s + 2.70·15-s + 4·17-s − 7.40·19-s + 21-s − 23-s + 2.29·25-s − 27-s + 6.70·29-s + 2·31-s − 4·33-s + 2.70·35-s + 10.7·37-s + 0.701·39-s − 6.70·41-s − 4.70·43-s − 2.70·45-s − 8.10·47-s + 49-s − 4·51-s + 3.40·53-s − 10.8·55-s + ⋯ |
| L(s) = 1 | − 0.577·3-s − 1.20·5-s − 0.377·7-s + 0.333·9-s + 1.20·11-s − 0.194·13-s + 0.697·15-s + 0.970·17-s − 1.69·19-s + 0.218·21-s − 0.208·23-s + 0.459·25-s − 0.192·27-s + 1.24·29-s + 0.359·31-s − 0.696·33-s + 0.456·35-s + 1.75·37-s + 0.112·39-s − 1.04·41-s − 0.716·43-s − 0.402·45-s − 1.18·47-s + 0.142·49-s − 0.560·51-s + 0.467·53-s − 1.45·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 7728 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7728 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.9404224530\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.9404224530\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + T \) |
| 7 | \( 1 + T \) |
| 23 | \( 1 + T \) |
| good | 5 | \( 1 + 2.70T + 5T^{2} \) |
| 11 | \( 1 - 4T + 11T^{2} \) |
| 13 | \( 1 + 0.701T + 13T^{2} \) |
| 17 | \( 1 - 4T + 17T^{2} \) |
| 19 | \( 1 + 7.40T + 19T^{2} \) |
| 29 | \( 1 - 6.70T + 29T^{2} \) |
| 31 | \( 1 - 2T + 31T^{2} \) |
| 37 | \( 1 - 10.7T + 37T^{2} \) |
| 41 | \( 1 + 6.70T + 41T^{2} \) |
| 43 | \( 1 + 4.70T + 43T^{2} \) |
| 47 | \( 1 + 8.10T + 47T^{2} \) |
| 53 | \( 1 - 3.40T + 53T^{2} \) |
| 59 | \( 1 + 5.40T + 59T^{2} \) |
| 61 | \( 1 + 2T + 61T^{2} \) |
| 67 | \( 1 - 10.8T + 67T^{2} \) |
| 71 | \( 1 + 5.40T + 71T^{2} \) |
| 73 | \( 1 + 11.4T + 73T^{2} \) |
| 79 | \( 1 + 8T + 79T^{2} \) |
| 83 | \( 1 + 0.596T + 83T^{2} \) |
| 89 | \( 1 - 1.40T + 89T^{2} \) |
| 97 | \( 1 - 12.7T + 97T^{2} \) |
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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
−7.930028048545582102393908123795, −7.05665042589200869385539318578, −6.46875349338535535528497497749, −6.00680814245760931563076776175, −4.85575657100300267313996839362, −4.28617306465315880877684609901, −3.71028616607640851484528745142, −2.85613960146247795396373140031, −1.56572353180311439580611930246, −0.51521887791342157925015831321,
0.51521887791342157925015831321, 1.56572353180311439580611930246, 2.85613960146247795396373140031, 3.71028616607640851484528745142, 4.28617306465315880877684609901, 4.85575657100300267313996839362, 6.00680814245760931563076776175, 6.46875349338535535528497497749, 7.05665042589200869385539318578, 7.930028048545582102393908123795
