L(s) = 1 | + (0.707 + 0.707i)5-s + i·7-s + (0.707 + 0.707i)11-s + (−1 − i)13-s + 1.41i·17-s − 31-s + (−0.707 + 0.707i)35-s + (1 − i)43-s + 1.41i·47-s + (−0.707 − 0.707i)53-s + 1.00i·55-s − 1.41i·65-s + (−1 − i)67-s + 1.41·71-s + i·73-s + ⋯ |
L(s) = 1 | + (0.707 + 0.707i)5-s + i·7-s + (0.707 + 0.707i)11-s + (−1 − i)13-s + 1.41i·17-s − 31-s + (−0.707 + 0.707i)35-s + (1 − i)43-s + 1.41i·47-s + (−0.707 − 0.707i)53-s + 1.00i·55-s − 1.41i·65-s + (−1 − i)67-s + 1.41·71-s + i·73-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1728 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.382 - 0.923i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1728 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.382 - 0.923i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(1.233905879\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.233905879\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
good | 5 | \( 1 + (-0.707 - 0.707i)T + iT^{2} \) |
| 7 | \( 1 - iT - T^{2} \) |
| 11 | \( 1 + (-0.707 - 0.707i)T + iT^{2} \) |
| 13 | \( 1 + (1 + i)T + iT^{2} \) |
| 17 | \( 1 - 1.41iT - T^{2} \) |
| 19 | \( 1 + iT^{2} \) |
| 23 | \( 1 + T^{2} \) |
| 29 | \( 1 - iT^{2} \) |
| 31 | \( 1 + T + T^{2} \) |
| 37 | \( 1 - iT^{2} \) |
| 41 | \( 1 + T^{2} \) |
| 43 | \( 1 + (-1 + i)T - iT^{2} \) |
| 47 | \( 1 - 1.41iT - T^{2} \) |
| 53 | \( 1 + (0.707 + 0.707i)T + iT^{2} \) |
| 59 | \( 1 + iT^{2} \) |
| 61 | \( 1 + iT^{2} \) |
| 67 | \( 1 + (1 + i)T + iT^{2} \) |
| 71 | \( 1 - 1.41T + T^{2} \) |
| 73 | \( 1 - iT - T^{2} \) |
| 79 | \( 1 + T^{2} \) |
| 83 | \( 1 + (-0.707 + 0.707i)T - iT^{2} \) |
| 89 | \( 1 - 1.41T + T^{2} \) |
| 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
−9.635267626356454775455233191156, −9.037972040293422263517966092073, −8.059338648484811005721100942922, −7.25707364344231969057443392337, −6.30731665863474286880736414604, −5.78136607915405100082449493527, −4.86704113590979641538307619655, −3.67198967629141611279574612688, −2.57643658410204539008691649513, −1.86249989552766588058839617478,
0.988546802046507834733748318804, 2.19137392873903803239592831973, 3.51119577129881542102389610905, 4.50149724821063543432988127956, 5.16747775415746864305776754321, 6.17304769351559242434115416871, 7.06455436619895528217560852122, 7.60678844659312842247690901736, 8.895893202914540090023633575803, 9.300933184979801461547204755441