L(s) = 1 | + (0.707 − 0.707i)2-s − 1.00i·4-s − 5-s + (0.707 + 0.707i)7-s + (−0.707 − 0.707i)8-s + (−0.707 + 0.707i)10-s − 1.41i·11-s + 1.00·14-s − 1.00·16-s + (1 − i)17-s − 1.41i·19-s + 1.00i·20-s + (−1.00 − 1.00i)22-s + 25-s + (0.707 − 0.707i)28-s + ⋯ |
L(s) = 1 | + (0.707 − 0.707i)2-s − 1.00i·4-s − 5-s + (0.707 + 0.707i)7-s + (−0.707 − 0.707i)8-s + (−0.707 + 0.707i)10-s − 1.41i·11-s + 1.00·14-s − 1.00·16-s + (1 − i)17-s − 1.41i·19-s + 1.00i·20-s + (−1.00 − 1.00i)22-s + 25-s + (0.707 − 0.707i)28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1260 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.229 + 0.973i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1260 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.229 + 0.973i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(1.337207566\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.337207566\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-0.707 + 0.707i)T \) |
| 3 | \( 1 \) |
| 5 | \( 1 + T \) |
| 7 | \( 1 + (-0.707 - 0.707i)T \) |
good | 11 | \( 1 + 1.41iT - T^{2} \) |
| 13 | \( 1 + iT^{2} \) |
| 17 | \( 1 + (-1 + i)T - iT^{2} \) |
| 19 | \( 1 + 1.41iT - T^{2} \) |
| 23 | \( 1 + iT^{2} \) |
| 29 | \( 1 - T^{2} \) |
| 31 | \( 1 + 1.41T + T^{2} \) |
| 37 | \( 1 + (-1 - i)T + iT^{2} \) |
| 41 | \( 1 - 2iT - T^{2} \) |
| 43 | \( 1 + iT^{2} \) |
| 47 | \( 1 + iT^{2} \) |
| 53 | \( 1 - iT^{2} \) |
| 59 | \( 1 - T^{2} \) |
| 61 | \( 1 - T^{2} \) |
| 67 | \( 1 - iT^{2} \) |
| 71 | \( 1 - 1.41iT - T^{2} \) |
| 73 | \( 1 + iT^{2} \) |
| 79 | \( 1 + T^{2} \) |
| 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
−9.683350743364153899078572234839, −8.890992832287390872452431187786, −8.142459364819612710232928877994, −7.16681271280985946135892853259, −6.05916156406297725524610447012, −5.19896686436612510562977372458, −4.52987198082840187326864923731, −3.31935060972493378996783704552, −2.72887767321862700831340790360, −1.01169715594222594540338677715,
1.90380789238496391701834953985, 3.66810844504495781289485995047, 4.01467021824935266083836555975, 4.97262316780492186515135190132, 5.86559465097898468830749126820, 7.07874672749192783409610740813, 7.61385779035667132795240750642, 8.030153026878363916304923860181, 9.068090236423037190830723535300, 10.26476029118101047528517954436