| L(s) = 1 | + 3-s − 4.22·5-s + 1.38·7-s + 9-s + 1.25·11-s − 0.245·13-s − 4.22·15-s + 1.32·17-s − 6.45·19-s + 1.38·21-s − 2.10·23-s + 12.8·25-s + 27-s − 1.84·29-s + 4.35·31-s + 1.25·33-s − 5.86·35-s + 9.22·37-s − 0.245·39-s − 7.82·41-s − 2.81·43-s − 4.22·45-s + 13.2·47-s − 5.07·49-s + 1.32·51-s − 12.9·53-s − 5.31·55-s + ⋯ |
| L(s) = 1 | + 0.577·3-s − 1.88·5-s + 0.524·7-s + 0.333·9-s + 0.379·11-s − 0.0680·13-s − 1.09·15-s + 0.322·17-s − 1.48·19-s + 0.302·21-s − 0.439·23-s + 2.56·25-s + 0.192·27-s − 0.343·29-s + 0.782·31-s + 0.219·33-s − 0.991·35-s + 1.51·37-s − 0.0392·39-s − 1.22·41-s − 0.429·43-s − 0.629·45-s + 1.92·47-s − 0.724·49-s + 0.186·51-s − 1.77·53-s − 0.717·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6024 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6024 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.516443509\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.516443509\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 - T \) |
| 251 | \( 1 + T \) |
| good | 5 | \( 1 + 4.22T + 5T^{2} \) |
| 7 | \( 1 - 1.38T + 7T^{2} \) |
| 11 | \( 1 - 1.25T + 11T^{2} \) |
| 13 | \( 1 + 0.245T + 13T^{2} \) |
| 17 | \( 1 - 1.32T + 17T^{2} \) |
| 19 | \( 1 + 6.45T + 19T^{2} \) |
| 23 | \( 1 + 2.10T + 23T^{2} \) |
| 29 | \( 1 + 1.84T + 29T^{2} \) |
| 31 | \( 1 - 4.35T + 31T^{2} \) |
| 37 | \( 1 - 9.22T + 37T^{2} \) |
| 41 | \( 1 + 7.82T + 41T^{2} \) |
| 43 | \( 1 + 2.81T + 43T^{2} \) |
| 47 | \( 1 - 13.2T + 47T^{2} \) |
| 53 | \( 1 + 12.9T + 53T^{2} \) |
| 59 | \( 1 + 13.5T + 59T^{2} \) |
| 61 | \( 1 - 7.85T + 61T^{2} \) |
| 67 | \( 1 - 6.41T + 67T^{2} \) |
| 71 | \( 1 - 8.37T + 71T^{2} \) |
| 73 | \( 1 - 0.162T + 73T^{2} \) |
| 79 | \( 1 + 7.17T + 79T^{2} \) |
| 83 | \( 1 - 3.36T + 83T^{2} \) |
| 89 | \( 1 + 9.35T + 89T^{2} \) |
| 97 | \( 1 - 11.1T + 97T^{2} \) |
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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
−8.131049188619425381115469033367, −7.59147049891886082197368390390, −6.85940506809519974251469017485, −6.10939085939548734282849718739, −4.78451717373477036622177178354, −4.38076427522296611248010998719, −3.70642422964905800569971849061, −2.97975584979178148774308163674, −1.88123323489667586267593088461, −0.62534812571293285721488922937,
0.62534812571293285721488922937, 1.88123323489667586267593088461, 2.97975584979178148774308163674, 3.70642422964905800569971849061, 4.38076427522296611248010998719, 4.78451717373477036622177178354, 6.10939085939548734282849718739, 6.85940506809519974251469017485, 7.59147049891886082197368390390, 8.131049188619425381115469033367
