| L(s) = 1 | − 3·3-s − 10.1·5-s + 7·7-s + 9·9-s + 36.1·11-s + 74.6·13-s + 30.4·15-s − 91.1·17-s + 104.·19-s − 21·21-s + 36.8·23-s − 21.6·25-s − 27·27-s + 262.·29-s − 310.·31-s − 108.·33-s − 71.1·35-s − 285.·37-s − 223.·39-s + 62.4·41-s − 386.·43-s − 91.4·45-s + 430.·47-s + 49·49-s + 273.·51-s − 111.·53-s − 367.·55-s + ⋯ |
| L(s) = 1 | − 0.577·3-s − 0.909·5-s + 0.377·7-s + 0.333·9-s + 0.991·11-s + 1.59·13-s + 0.524·15-s − 1.30·17-s + 1.25·19-s − 0.218·21-s + 0.333·23-s − 0.173·25-s − 0.192·27-s + 1.68·29-s − 1.80·31-s − 0.572·33-s − 0.343·35-s − 1.26·37-s − 0.919·39-s + 0.238·41-s − 1.37·43-s − 0.303·45-s + 1.33·47-s + 0.142·49-s + 0.750·51-s − 0.290·53-s − 0.901·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1344 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1344 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(1.625572267\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.625572267\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + 3T \) |
| 7 | \( 1 - 7T \) |
| good | 5 | \( 1 + 10.1T + 125T^{2} \) |
| 11 | \( 1 - 36.1T + 1.33e3T^{2} \) |
| 13 | \( 1 - 74.6T + 2.19e3T^{2} \) |
| 17 | \( 1 + 91.1T + 4.91e3T^{2} \) |
| 19 | \( 1 - 104.T + 6.85e3T^{2} \) |
| 23 | \( 1 - 36.8T + 1.21e4T^{2} \) |
| 29 | \( 1 - 262.T + 2.43e4T^{2} \) |
| 31 | \( 1 + 310.T + 2.97e4T^{2} \) |
| 37 | \( 1 + 285.T + 5.06e4T^{2} \) |
| 41 | \( 1 - 62.4T + 6.89e4T^{2} \) |
| 43 | \( 1 + 386.T + 7.95e4T^{2} \) |
| 47 | \( 1 - 430.T + 1.03e5T^{2} \) |
| 53 | \( 1 + 111.T + 1.48e5T^{2} \) |
| 59 | \( 1 - 479.T + 2.05e5T^{2} \) |
| 61 | \( 1 - 602.T + 2.26e5T^{2} \) |
| 67 | \( 1 + 1.02e3T + 3.00e5T^{2} \) |
| 71 | \( 1 - 284.T + 3.57e5T^{2} \) |
| 73 | \( 1 + 566.T + 3.89e5T^{2} \) |
| 79 | \( 1 + 94.2T + 4.93e5T^{2} \) |
| 83 | \( 1 + 626.T + 5.71e5T^{2} \) |
| 89 | \( 1 - 1.53e3T + 7.04e5T^{2} \) |
| 97 | \( 1 - 718.T + 9.12e5T^{2} \) |
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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
−8.927085929046321809956834034309, −8.647165209288432985495921943962, −7.48844109939000386831411637946, −6.81009789542234815137926661860, −5.98838147800676108479801049132, −4.97178631679906150373572917469, −4.04613382647138234455809879799, −3.40931493147792888682224917621, −1.68866682764778090114613250937, −0.69425843033233957050355496860,
0.69425843033233957050355496860, 1.68866682764778090114613250937, 3.40931493147792888682224917621, 4.04613382647138234455809879799, 4.97178631679906150373572917469, 5.98838147800676108479801049132, 6.81009789542234815137926661860, 7.48844109939000386831411637946, 8.647165209288432985495921943962, 8.927085929046321809956834034309
