| L(s) = 1 | + 2.79·3-s + 5-s + 4.28·7-s + 4.80·9-s − 3.82·11-s − 1.37·13-s + 2.79·15-s + 1.08·17-s + 1.88·19-s + 11.9·21-s + 0.393·23-s + 25-s + 5.03·27-s + 4.59·29-s + 2.75·31-s − 10.6·33-s + 4.28·35-s − 6.07·37-s − 3.82·39-s + 1.05·41-s + 7.29·43-s + 4.80·45-s − 1.97·47-s + 11.3·49-s + 3.03·51-s + 3.67·53-s − 3.82·55-s + ⋯ |
| L(s) = 1 | + 1.61·3-s + 0.447·5-s + 1.61·7-s + 1.60·9-s − 1.15·11-s − 0.380·13-s + 0.721·15-s + 0.263·17-s + 0.433·19-s + 2.61·21-s + 0.0821·23-s + 0.200·25-s + 0.968·27-s + 0.852·29-s + 0.494·31-s − 1.86·33-s + 0.724·35-s − 0.998·37-s − 0.612·39-s + 0.164·41-s + 1.11·43-s + 0.715·45-s − 0.288·47-s + 1.62·49-s + 0.424·51-s + 0.505·53-s − 0.515·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6040 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6040 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(4.899101728\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.899101728\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 - T \) |
| 151 | \( 1 + T \) |
| good | 3 | \( 1 - 2.79T + 3T^{2} \) |
| 7 | \( 1 - 4.28T + 7T^{2} \) |
| 11 | \( 1 + 3.82T + 11T^{2} \) |
| 13 | \( 1 + 1.37T + 13T^{2} \) |
| 17 | \( 1 - 1.08T + 17T^{2} \) |
| 19 | \( 1 - 1.88T + 19T^{2} \) |
| 23 | \( 1 - 0.393T + 23T^{2} \) |
| 29 | \( 1 - 4.59T + 29T^{2} \) |
| 31 | \( 1 - 2.75T + 31T^{2} \) |
| 37 | \( 1 + 6.07T + 37T^{2} \) |
| 41 | \( 1 - 1.05T + 41T^{2} \) |
| 43 | \( 1 - 7.29T + 43T^{2} \) |
| 47 | \( 1 + 1.97T + 47T^{2} \) |
| 53 | \( 1 - 3.67T + 53T^{2} \) |
| 59 | \( 1 - 8.88T + 59T^{2} \) |
| 61 | \( 1 + 5.24T + 61T^{2} \) |
| 67 | \( 1 + 5.54T + 67T^{2} \) |
| 71 | \( 1 - 2.99T + 71T^{2} \) |
| 73 | \( 1 + 1.44T + 73T^{2} \) |
| 79 | \( 1 + 6.78T + 79T^{2} \) |
| 83 | \( 1 + 2.39T + 83T^{2} \) |
| 89 | \( 1 - 14.5T + 89T^{2} \) |
| 97 | \( 1 - 1.16T + 97T^{2} \) |
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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
−8.035475508974504823883549908283, −7.67170624127056305451513840649, −7.03903555543771062735456210158, −5.78983889663700285459307455617, −5.00982304797191091001960641674, −4.50246498253682713740835055921, −3.45513847553013980886784335754, −2.59389725031444693221768743665, −2.11550782174233241638487447239, −1.16891864012084012385345752641,
1.16891864012084012385345752641, 2.11550782174233241638487447239, 2.59389725031444693221768743665, 3.45513847553013980886784335754, 4.50246498253682713740835055921, 5.00982304797191091001960641674, 5.78983889663700285459307455617, 7.03903555543771062735456210158, 7.67170624127056305451513840649, 8.035475508974504823883549908283
