| L(s) = 1 | − 1.15·3-s + 4.11·5-s + 1.30·7-s − 1.66·9-s − 11-s − 6.30·13-s − 4.74·15-s − 3.49·17-s + 7.02·19-s − 1.50·21-s + 4.30·23-s + 11.9·25-s + 5.38·27-s + 7.31·29-s + 4.67·31-s + 1.15·33-s + 5.38·35-s − 7.08·37-s + 7.26·39-s − 5.46·41-s + 0.276·43-s − 6.86·45-s − 2.61·47-s − 5.28·49-s + 4.02·51-s + 53-s − 4.11·55-s + ⋯ |
| L(s) = 1 | − 0.666·3-s + 1.84·5-s + 0.494·7-s − 0.556·9-s − 0.301·11-s − 1.74·13-s − 1.22·15-s − 0.846·17-s + 1.61·19-s − 0.329·21-s + 0.898·23-s + 2.38·25-s + 1.03·27-s + 1.35·29-s + 0.840·31-s + 0.200·33-s + 0.910·35-s − 1.16·37-s + 1.16·39-s − 0.852·41-s + 0.0421·43-s − 1.02·45-s − 0.381·47-s − 0.755·49-s + 0.564·51-s + 0.137·53-s − 0.554·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 9328 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 9328 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.184631014\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.184631014\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 11 | \( 1 + T \) |
| 53 | \( 1 - T \) |
| good | 3 | \( 1 + 1.15T + 3T^{2} \) |
| 5 | \( 1 - 4.11T + 5T^{2} \) |
| 7 | \( 1 - 1.30T + 7T^{2} \) |
| 13 | \( 1 + 6.30T + 13T^{2} \) |
| 17 | \( 1 + 3.49T + 17T^{2} \) |
| 19 | \( 1 - 7.02T + 19T^{2} \) |
| 23 | \( 1 - 4.30T + 23T^{2} \) |
| 29 | \( 1 - 7.31T + 29T^{2} \) |
| 31 | \( 1 - 4.67T + 31T^{2} \) |
| 37 | \( 1 + 7.08T + 37T^{2} \) |
| 41 | \( 1 + 5.46T + 41T^{2} \) |
| 43 | \( 1 - 0.276T + 43T^{2} \) |
| 47 | \( 1 + 2.61T + 47T^{2} \) |
| 59 | \( 1 + 6.72T + 59T^{2} \) |
| 61 | \( 1 - 6.90T + 61T^{2} \) |
| 67 | \( 1 + 6.91T + 67T^{2} \) |
| 71 | \( 1 - 11.0T + 71T^{2} \) |
| 73 | \( 1 - 8.79T + 73T^{2} \) |
| 79 | \( 1 - 5.94T + 79T^{2} \) |
| 83 | \( 1 - 11.4T + 83T^{2} \) |
| 89 | \( 1 - 1.99T + 89T^{2} \) |
| 97 | \( 1 + 5.08T + 97T^{2} \) |
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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
−7.60161244619430105175829136092, −6.69745264775834480698407492319, −6.46437254018611955617826755959, −5.41600415422165583564487554952, −5.03193756326646379787339034041, −4.84287935176958921085008262710, −3.07530664012282869764915206263, −2.57951561603752978363962200929, −1.76918181701838400890366571362, −0.73357587469024963797136010703,
0.73357587469024963797136010703, 1.76918181701838400890366571362, 2.57951561603752978363962200929, 3.07530664012282869764915206263, 4.84287935176958921085008262710, 5.03193756326646379787339034041, 5.41600415422165583564487554952, 6.46437254018611955617826755959, 6.69745264775834480698407492319, 7.60161244619430105175829136092
