| L(s) = 1 | − 1.87·3-s + 0.347·5-s − 4.75·7-s + 0.532·9-s + 4.75·11-s + 1.87·13-s − 0.652·15-s + 1.53·17-s + 8.94·21-s − 6.10·23-s − 4.87·25-s + 4.63·27-s + 1.98·29-s + 3.36·31-s − 8.94·33-s − 1.65·35-s + 3.38·37-s − 3.53·39-s + 4.49·41-s + 2.10·43-s + 0.184·45-s − 4.92·47-s + 15.6·49-s − 2.87·51-s + 12.5·53-s + 1.65·55-s − 5.66·59-s + ⋯ |
| L(s) = 1 | − 1.08·3-s + 0.155·5-s − 1.79·7-s + 0.177·9-s + 1.43·11-s + 0.521·13-s − 0.168·15-s + 0.371·17-s + 1.95·21-s − 1.27·23-s − 0.975·25-s + 0.892·27-s + 0.368·29-s + 0.605·31-s − 1.55·33-s − 0.279·35-s + 0.557·37-s − 0.565·39-s + 0.701·41-s + 0.321·43-s + 0.0275·45-s − 0.717·47-s + 2.23·49-s − 0.403·51-s + 1.72·53-s + 0.222·55-s − 0.736·59-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2888 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2888 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 19 | \( 1 \) |
| good | 3 | \( 1 + 1.87T + 3T^{2} \) |
| 5 | \( 1 - 0.347T + 5T^{2} \) |
| 7 | \( 1 + 4.75T + 7T^{2} \) |
| 11 | \( 1 - 4.75T + 11T^{2} \) |
| 13 | \( 1 - 1.87T + 13T^{2} \) |
| 17 | \( 1 - 1.53T + 17T^{2} \) |
| 23 | \( 1 + 6.10T + 23T^{2} \) |
| 29 | \( 1 - 1.98T + 29T^{2} \) |
| 31 | \( 1 - 3.36T + 31T^{2} \) |
| 37 | \( 1 - 3.38T + 37T^{2} \) |
| 41 | \( 1 - 4.49T + 41T^{2} \) |
| 43 | \( 1 - 2.10T + 43T^{2} \) |
| 47 | \( 1 + 4.92T + 47T^{2} \) |
| 53 | \( 1 - 12.5T + 53T^{2} \) |
| 59 | \( 1 + 5.66T + 59T^{2} \) |
| 61 | \( 1 - 9.99T + 61T^{2} \) |
| 67 | \( 1 + 10.5T + 67T^{2} \) |
| 71 | \( 1 + 11.4T + 71T^{2} \) |
| 73 | \( 1 + 14.6T + 73T^{2} \) |
| 79 | \( 1 + 2.36T + 79T^{2} \) |
| 83 | \( 1 + 11.5T + 83T^{2} \) |
| 89 | \( 1 + 5.63T + 89T^{2} \) |
| 97 | \( 1 + 7.66T + 97T^{2} \) |
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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
−8.578421672744156088252051352475, −7.37303248041090200257910981749, −6.57428916355476740976062191620, −6.00176020919290467996082861047, −5.81239313682952076637644364775, −4.33595985182479831217567862412, −3.71554757525713149617666034404, −2.71399180991414217572485245461, −1.18792361312193777263720358567, 0,
1.18792361312193777263720358567, 2.71399180991414217572485245461, 3.71554757525713149617666034404, 4.33595985182479831217567862412, 5.81239313682952076637644364775, 6.00176020919290467996082861047, 6.57428916355476740976062191620, 7.37303248041090200257910981749, 8.578421672744156088252051352475
