L(s) = 1 | + (0.258 + 0.965i)2-s + (−0.866 + 0.499i)4-s + (0.707 − 0.707i)7-s + (−0.707 − 0.707i)8-s + i·9-s + 1.73i·11-s + (0.866 + 0.500i)14-s + (0.500 − 0.866i)16-s + (−0.965 + 0.258i)18-s + (−1.67 + 0.448i)22-s + (0.707 + 0.707i)23-s + (−0.258 + 0.965i)28-s − i·29-s + (0.965 + 0.258i)32-s + (−0.499 − 0.866i)36-s + (−1.22 − 1.22i)37-s + ⋯ |
L(s) = 1 | + (0.258 + 0.965i)2-s + (−0.866 + 0.499i)4-s + (0.707 − 0.707i)7-s + (−0.707 − 0.707i)8-s + i·9-s + 1.73i·11-s + (0.866 + 0.500i)14-s + (0.500 − 0.866i)16-s + (−0.965 + 0.258i)18-s + (−1.67 + 0.448i)22-s + (0.707 + 0.707i)23-s + (−0.258 + 0.965i)28-s − i·29-s + (0.965 + 0.258i)32-s + (−0.499 − 0.866i)36-s + (−1.22 − 1.22i)37-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 700 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.189 - 0.981i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 700 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.189 - 0.981i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(1.021141299\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.021141299\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-0.258 - 0.965i)T \) |
| 5 | \( 1 \) |
| 7 | \( 1 + (-0.707 + 0.707i)T \) |
good | 3 | \( 1 - iT^{2} \) |
| 11 | \( 1 - 1.73iT - T^{2} \) |
| 13 | \( 1 + iT^{2} \) |
| 17 | \( 1 - iT^{2} \) |
| 19 | \( 1 - T^{2} \) |
| 23 | \( 1 + (-0.707 - 0.707i)T + iT^{2} \) |
| 29 | \( 1 + iT - T^{2} \) |
| 31 | \( 1 + T^{2} \) |
| 37 | \( 1 + (1.22 + 1.22i)T + iT^{2} \) |
| 41 | \( 1 - T^{2} \) |
| 43 | \( 1 + (0.707 + 0.707i)T + iT^{2} \) |
| 47 | \( 1 + iT^{2} \) |
| 53 | \( 1 - iT^{2} \) |
| 59 | \( 1 - T^{2} \) |
| 61 | \( 1 - T^{2} \) |
| 67 | \( 1 + (-0.707 + 0.707i)T - iT^{2} \) |
| 71 | \( 1 + 1.73iT - T^{2} \) |
| 73 | \( 1 + iT^{2} \) |
| 79 | \( 1 - 1.73T + T^{2} \) |
| 83 | \( 1 - iT^{2} \) |
| 89 | \( 1 + T^{2} \) |
| 97 | \( 1 - iT^{2} \) |
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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
−10.75211599348353863405258966867, −9.964647190554390981696782640025, −9.054392002372552985228711573342, −7.906552358505447036317718354883, −7.47684906538455315705803426497, −6.71976293802846803509204039572, −5.27610741289750045750402563679, −4.76782229060643164186192278365, −3.81019901284444562517276017224, −1.99890243323828732158319712101,
1.21666096609023860472941152110, 2.81768883972257546820327323100, 3.59351526258679245455076876853, 4.90036714821025696929582212521, 5.71383856550288855664704992372, 6.63841932486733731973910769972, 8.424465113657670869833415313398, 8.641963041636095553842917747629, 9.591093193114020833478290211755, 10.68027621927717739353328126882