| L(s) = 1 | − 0.414·2-s + 2·3-s − 1.82·4-s + 5-s − 0.828·6-s + 2.82·7-s + 1.58·8-s + 9-s − 0.414·10-s − 11-s − 3.65·12-s + 1.17·13-s − 1.17·14-s + 2·15-s + 3·16-s + 0.828·17-s − 0.414·18-s − 19-s − 1.82·20-s + 5.65·21-s + 0.414·22-s + 4·23-s + 3.17·24-s + 25-s − 0.485·26-s − 4·27-s − 5.17·28-s + ⋯ |
| L(s) = 1 | − 0.292·2-s + 1.15·3-s − 0.914·4-s + 0.447·5-s − 0.338·6-s + 1.06·7-s + 0.560·8-s + 0.333·9-s − 0.130·10-s − 0.301·11-s − 1.05·12-s + 0.324·13-s − 0.313·14-s + 0.516·15-s + 0.750·16-s + 0.200·17-s − 0.0976·18-s − 0.229·19-s − 0.408·20-s + 1.23·21-s + 0.0883·22-s + 0.834·23-s + 0.647·24-s + 0.200·25-s − 0.0951·26-s − 0.769·27-s − 0.977·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1045 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1045 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.038384450\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.038384450\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 - T \) |
| 11 | \( 1 + T \) |
| 19 | \( 1 + T \) |
| good | 2 | \( 1 + 0.414T + 2T^{2} \) |
| 3 | \( 1 - 2T + 3T^{2} \) |
| 7 | \( 1 - 2.82T + 7T^{2} \) |
| 13 | \( 1 - 1.17T + 13T^{2} \) |
| 17 | \( 1 - 0.828T + 17T^{2} \) |
| 23 | \( 1 - 4T + 23T^{2} \) |
| 29 | \( 1 - 8.82T + 29T^{2} \) |
| 31 | \( 1 - 6.82T + 31T^{2} \) |
| 37 | \( 1 + 6.82T + 37T^{2} \) |
| 41 | \( 1 + 3.17T + 41T^{2} \) |
| 43 | \( 1 + 1.17T + 43T^{2} \) |
| 47 | \( 1 + 47T^{2} \) |
| 53 | \( 1 - 2.82T + 53T^{2} \) |
| 59 | \( 1 + 4.48T + 59T^{2} \) |
| 61 | \( 1 - 7.65T + 61T^{2} \) |
| 67 | \( 1 - 11.6T + 67T^{2} \) |
| 71 | \( 1 - 14.8T + 71T^{2} \) |
| 73 | \( 1 - 6.48T + 73T^{2} \) |
| 79 | \( 1 + 12T + 79T^{2} \) |
| 83 | \( 1 + 2.82T + 83T^{2} \) |
| 89 | \( 1 - 9.31T + 89T^{2} \) |
| 97 | \( 1 + 18.1T + 97T^{2} \) |
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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
−9.785512387266771973485987118759, −8.859645511320313339269777859808, −8.372993657359898213658539994643, −7.931134464477924215164870637458, −6.72867725196154039860846369462, −5.35185830673477984120739825870, −4.69467750720940404804277596242, −3.58569590515414638411341916070, −2.47803105698748227074678920147, −1.23022502596973767761813067277,
1.23022502596973767761813067277, 2.47803105698748227074678920147, 3.58569590515414638411341916070, 4.69467750720940404804277596242, 5.35185830673477984120739825870, 6.72867725196154039860846369462, 7.931134464477924215164870637458, 8.372993657359898213658539994643, 8.859645511320313339269777859808, 9.785512387266771973485987118759
