| L(s) = 1 | + 2.87·3-s + 2.53·5-s − 7-s + 5.29·9-s + 2.83·11-s − 2.18·13-s + 7.29·15-s + 0.411·17-s + 6.63·19-s − 2.87·21-s + 7.75·23-s + 1.41·25-s + 6.59·27-s − 9.98·29-s − 3.45·31-s + 8.17·33-s − 2.53·35-s − 2.04·37-s − 6.29·39-s + 4.29·41-s + 7.68·43-s + 13.3·45-s − 11.4·47-s + 49-s + 1.18·51-s + 6.10·53-s + 7.18·55-s + ⋯ |
| L(s) = 1 | + 1.66·3-s + 1.13·5-s − 0.377·7-s + 1.76·9-s + 0.855·11-s − 0.605·13-s + 1.88·15-s + 0.0997·17-s + 1.52·19-s − 0.628·21-s + 1.61·23-s + 0.282·25-s + 1.26·27-s − 1.85·29-s − 0.620·31-s + 1.42·33-s − 0.428·35-s − 0.335·37-s − 1.00·39-s + 0.670·41-s + 1.17·43-s + 1.99·45-s − 1.66·47-s + 0.142·49-s + 0.165·51-s + 0.838·53-s + 0.968·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 7504 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7504 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(5.286134209\) |
| \(L(\frac12)\) |
\(\approx\) |
\(5.286134209\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 7 | \( 1 + T \) |
| 67 | \( 1 + T \) |
| good | 3 | \( 1 - 2.87T + 3T^{2} \) |
| 5 | \( 1 - 2.53T + 5T^{2} \) |
| 11 | \( 1 - 2.83T + 11T^{2} \) |
| 13 | \( 1 + 2.18T + 13T^{2} \) |
| 17 | \( 1 - 0.411T + 17T^{2} \) |
| 19 | \( 1 - 6.63T + 19T^{2} \) |
| 23 | \( 1 - 7.75T + 23T^{2} \) |
| 29 | \( 1 + 9.98T + 29T^{2} \) |
| 31 | \( 1 + 3.45T + 31T^{2} \) |
| 37 | \( 1 + 2.04T + 37T^{2} \) |
| 41 | \( 1 - 4.29T + 41T^{2} \) |
| 43 | \( 1 - 7.68T + 43T^{2} \) |
| 47 | \( 1 + 11.4T + 47T^{2} \) |
| 53 | \( 1 - 6.10T + 53T^{2} \) |
| 59 | \( 1 - 11.5T + 59T^{2} \) |
| 61 | \( 1 - 3.04T + 61T^{2} \) |
| 71 | \( 1 - 13.3T + 71T^{2} \) |
| 73 | \( 1 + 10.0T + 73T^{2} \) |
| 79 | \( 1 + 2.82T + 79T^{2} \) |
| 83 | \( 1 + 9.30T + 83T^{2} \) |
| 89 | \( 1 - 15.0T + 89T^{2} \) |
| 97 | \( 1 + 0.638T + 97T^{2} \) |
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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
−7.84458212056317717304230399601, −7.24340240551849542084106266436, −6.78837914346669995716640986779, −5.69685362912585645907162672768, −5.16783002476112819727305765937, −4.00956665248827700765718700831, −3.39565643331367103557141145698, −2.69409663992458582759769502925, −1.96451219720985186715964425758, −1.16133608478837941955737087680,
1.16133608478837941955737087680, 1.96451219720985186715964425758, 2.69409663992458582759769502925, 3.39565643331367103557141145698, 4.00956665248827700765718700831, 5.16783002476112819727305765937, 5.69685362912585645907162672768, 6.78837914346669995716640986779, 7.24340240551849542084106266436, 7.84458212056317717304230399601
