L(s) = 1 | − 2-s − 3-s + 4-s − 2.05·5-s + 6-s + 5.03·7-s − 8-s + 9-s + 2.05·10-s + 3.25·11-s − 12-s + 13-s − 5.03·14-s + 2.05·15-s + 16-s + 0.479·17-s − 18-s + 5.58·19-s − 2.05·20-s − 5.03·21-s − 3.25·22-s − 6.13·23-s + 24-s − 0.793·25-s − 26-s − 27-s + 5.03·28-s + ⋯ |
L(s) = 1 | − 0.707·2-s − 0.577·3-s + 0.5·4-s − 0.917·5-s + 0.408·6-s + 1.90·7-s − 0.353·8-s + 0.333·9-s + 0.648·10-s + 0.981·11-s − 0.288·12-s + 0.277·13-s − 1.34·14-s + 0.529·15-s + 0.250·16-s + 0.116·17-s − 0.235·18-s + 1.28·19-s − 0.458·20-s − 1.09·21-s − 0.694·22-s − 1.27·23-s + 0.204·24-s − 0.158·25-s − 0.196·26-s − 0.192·27-s + 0.951·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8034 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8034 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.545843010\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.545843010\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + T \) |
| 3 | \( 1 + T \) |
| 13 | \( 1 - T \) |
| 103 | \( 1 + T \) |
good | 5 | \( 1 + 2.05T + 5T^{2} \) |
| 7 | \( 1 - 5.03T + 7T^{2} \) |
| 11 | \( 1 - 3.25T + 11T^{2} \) |
| 17 | \( 1 - 0.479T + 17T^{2} \) |
| 19 | \( 1 - 5.58T + 19T^{2} \) |
| 23 | \( 1 + 6.13T + 23T^{2} \) |
| 29 | \( 1 - 3.21T + 29T^{2} \) |
| 31 | \( 1 - 8.30T + 31T^{2} \) |
| 37 | \( 1 + 3.88T + 37T^{2} \) |
| 41 | \( 1 - 5.27T + 41T^{2} \) |
| 43 | \( 1 - 3.12T + 43T^{2} \) |
| 47 | \( 1 - 3.55T + 47T^{2} \) |
| 53 | \( 1 + 6.76T + 53T^{2} \) |
| 59 | \( 1 - 13.1T + 59T^{2} \) |
| 61 | \( 1 - 6.84T + 61T^{2} \) |
| 67 | \( 1 + 3.51T + 67T^{2} \) |
| 71 | \( 1 - 0.616T + 71T^{2} \) |
| 73 | \( 1 - 13.2T + 73T^{2} \) |
| 79 | \( 1 - 2.05T + 79T^{2} \) |
| 83 | \( 1 + 2.18T + 83T^{2} \) |
| 89 | \( 1 + 3.35T + 89T^{2} \) |
| 97 | \( 1 + 1.04T + 97T^{2} \) |
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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
−7.988893472802547673960217244823, −7.34806000649957420351728852068, −6.60345633296678568297676456750, −5.76035195685537948002892881790, −5.04761735888372859756699481659, −4.25483575044017747101110045088, −3.73776453516265249758211201268, −2.40119216391729966816659620026, −1.39999677892357748464225810833, −0.811643889722881371981280178321,
0.811643889722881371981280178321, 1.39999677892357748464225810833, 2.40119216391729966816659620026, 3.73776453516265249758211201268, 4.25483575044017747101110045088, 5.04761735888372859756699481659, 5.76035195685537948002892881790, 6.60345633296678568297676456750, 7.34806000649957420351728852068, 7.988893472802547673960217244823