| L(s) = 1 | − 3-s + 0.926·5-s − 7-s + 9-s − 4.21·11-s + 13-s − 0.926·15-s − 2.87·17-s + 1.28·19-s + 21-s + 8.02·23-s − 4.14·25-s − 27-s + 3.28·29-s + 7.04·31-s + 4.21·33-s − 0.926·35-s + 8.57·37-s − 39-s − 12.0·41-s − 7.14·43-s + 0.926·45-s − 1.95·47-s + 49-s + 2.87·51-s − 5.14·53-s − 3.90·55-s + ⋯ |
| L(s) = 1 | − 0.577·3-s + 0.414·5-s − 0.377·7-s + 0.333·9-s − 1.27·11-s + 0.277·13-s − 0.239·15-s − 0.698·17-s + 0.295·19-s + 0.218·21-s + 1.67·23-s − 0.828·25-s − 0.192·27-s + 0.610·29-s + 1.26·31-s + 0.733·33-s − 0.156·35-s + 1.40·37-s − 0.160·39-s − 1.88·41-s − 1.08·43-s + 0.138·45-s − 0.284·47-s + 0.142·49-s + 0.403·51-s − 0.706·53-s − 0.526·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4368 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4368 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + T \) |
| 13 | \( 1 - T \) |
| good | 5 | \( 1 - 0.926T + 5T^{2} \) |
| 11 | \( 1 + 4.21T + 11T^{2} \) |
| 17 | \( 1 + 2.87T + 17T^{2} \) |
| 19 | \( 1 - 1.28T + 19T^{2} \) |
| 23 | \( 1 - 8.02T + 23T^{2} \) |
| 29 | \( 1 - 3.28T + 29T^{2} \) |
| 31 | \( 1 - 7.04T + 31T^{2} \) |
| 37 | \( 1 - 8.57T + 37T^{2} \) |
| 41 | \( 1 + 12.0T + 41T^{2} \) |
| 43 | \( 1 + 7.14T + 43T^{2} \) |
| 47 | \( 1 + 1.95T + 47T^{2} \) |
| 53 | \( 1 + 5.14T + 53T^{2} \) |
| 59 | \( 1 - 7.33T + 59T^{2} \) |
| 61 | \( 1 - 7.75T + 61T^{2} \) |
| 67 | \( 1 + 12.0T + 67T^{2} \) |
| 71 | \( 1 + 10.7T + 71T^{2} \) |
| 73 | \( 1 + 8.32T + 73T^{2} \) |
| 79 | \( 1 + 4.47T + 79T^{2} \) |
| 83 | \( 1 - 3.80T + 83T^{2} \) |
| 89 | \( 1 + 5.64T + 89T^{2} \) |
| 97 | \( 1 - 6.90T + 97T^{2} \) |
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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
−8.044164987188879506559485464946, −7.10922213657091929661259496445, −6.53434125121001600912769158801, −5.78638959430261682296768181874, −5.06541436665969022111322355608, −4.46295398681368225132995224340, −3.21462714442190887873889488905, −2.52541767103841029529796593057, −1.29374911486001396090487841606, 0,
1.29374911486001396090487841606, 2.52541767103841029529796593057, 3.21462714442190887873889488905, 4.46295398681368225132995224340, 5.06541436665969022111322355608, 5.78638959430261682296768181874, 6.53434125121001600912769158801, 7.10922213657091929661259496445, 8.044164987188879506559485464946
