L(s) = 1 | + 7-s + 1.41i·11-s + 13-s + 1.41i·17-s − 19-s + 1.41i·23-s − 1.41i·29-s − 31-s − 43-s + 1.41i·47-s − 1.41i·59-s + 61-s + 67-s + 1.41i·77-s − 1.41i·83-s + ⋯ |
L(s) = 1 | + 7-s + 1.41i·11-s + 13-s + 1.41i·17-s − 19-s + 1.41i·23-s − 1.41i·29-s − 31-s − 43-s + 1.41i·47-s − 1.41i·59-s + 61-s + 67-s + 1.41i·77-s − 1.41i·83-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3600 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.577 - 0.816i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3600 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.577 - 0.816i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(1.400968745\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.400968745\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 5 | \( 1 \) |
good | 7 | \( 1 - T + T^{2} \) |
| 11 | \( 1 - 1.41iT - T^{2} \) |
| 13 | \( 1 - T + T^{2} \) |
| 17 | \( 1 - 1.41iT - T^{2} \) |
| 19 | \( 1 + T + T^{2} \) |
| 23 | \( 1 - 1.41iT - T^{2} \) |
| 29 | \( 1 + 1.41iT - T^{2} \) |
| 31 | \( 1 + T + T^{2} \) |
| 37 | \( 1 + T^{2} \) |
| 41 | \( 1 - T^{2} \) |
| 43 | \( 1 + T + T^{2} \) |
| 47 | \( 1 - 1.41iT - T^{2} \) |
| 53 | \( 1 - T^{2} \) |
| 59 | \( 1 + 1.41iT - T^{2} \) |
| 61 | \( 1 - T + T^{2} \) |
| 67 | \( 1 - T + T^{2} \) |
| 71 | \( 1 - T^{2} \) |
| 73 | \( 1 + T^{2} \) |
| 79 | \( 1 + T^{2} \) |
| 83 | \( 1 + 1.41iT - T^{2} \) |
| 89 | \( 1 - T^{2} \) |
| 97 | \( 1 - T + T^{2} \) |
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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
−8.710564308136195994525248405738, −8.055589366567735307324079009058, −7.55525502599468323412494881251, −6.56126826123472603347650751197, −5.90005240869906844789575128064, −4.98561557376672697548185602270, −4.22718171592799916570217338214, −3.58369756359856413017064034223, −2.02664624170740135540600264529, −1.61245723757089528019772218303,
0.854814433917733532252852419903, 2.06006397745463030029900780347, 3.12468352684651830307925912594, 3.94764311166269882958010416617, 4.93607296628208658586609165469, 5.50673453653664649039742821198, 6.44428374035513674183502844617, 7.06957455539179760687525647156, 8.112920924487967028762584241507, 8.638500256103326820405106902297