L(s) = 1 | + (−0.278 − 0.142i)3-s + (0.951 − 0.309i)4-s + i·5-s + (−0.530 − 0.729i)9-s + (−0.309 − 0.0489i)12-s + (0.142 − 0.278i)15-s + (0.809 − 0.587i)16-s + (0.309 + 0.951i)20-s + (1.76 − 0.278i)23-s − 25-s + (0.0930 + 0.587i)27-s + (0.363 − 1.11i)31-s + (−0.729 − 0.530i)36-s + (1.39 + 0.221i)37-s + (0.729 − 0.530i)45-s + ⋯ |
L(s) = 1 | + (−0.278 − 0.142i)3-s + (0.951 − 0.309i)4-s + i·5-s + (−0.530 − 0.729i)9-s + (−0.309 − 0.0489i)12-s + (0.142 − 0.278i)15-s + (0.809 − 0.587i)16-s + (0.309 + 0.951i)20-s + (1.76 − 0.278i)23-s − 25-s + (0.0930 + 0.587i)27-s + (0.363 − 1.11i)31-s + (−0.729 − 0.530i)36-s + (1.39 + 0.221i)37-s + (0.729 − 0.530i)45-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3025 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.992 + 0.125i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3025 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.992 + 0.125i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(1.426775448\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.426775448\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 5 | \( 1 - iT \) |
| 11 | \( 1 \) |
good | 2 | \( 1 + (-0.951 + 0.309i)T^{2} \) |
| 3 | \( 1 + (0.278 + 0.142i)T + (0.587 + 0.809i)T^{2} \) |
| 7 | \( 1 + iT^{2} \) |
| 13 | \( 1 + (-0.951 - 0.309i)T^{2} \) |
| 17 | \( 1 + (0.587 - 0.809i)T^{2} \) |
| 19 | \( 1 + (0.809 + 0.587i)T^{2} \) |
| 23 | \( 1 + (-1.76 + 0.278i)T + (0.951 - 0.309i)T^{2} \) |
| 29 | \( 1 + (0.809 - 0.587i)T^{2} \) |
| 31 | \( 1 + (-0.363 + 1.11i)T + (-0.809 - 0.587i)T^{2} \) |
| 37 | \( 1 + (-1.39 - 0.221i)T + (0.951 + 0.309i)T^{2} \) |
| 41 | \( 1 + (0.309 - 0.951i)T^{2} \) |
| 43 | \( 1 - iT^{2} \) |
| 47 | \( 1 + (0.642 - 1.26i)T + (-0.587 - 0.809i)T^{2} \) |
| 53 | \( 1 + (-1.76 - 0.896i)T + (0.587 + 0.809i)T^{2} \) |
| 59 | \( 1 + (-0.690 - 0.951i)T + (-0.309 + 0.951i)T^{2} \) |
| 61 | \( 1 + (0.309 + 0.951i)T^{2} \) |
| 67 | \( 1 + (0.809 - 0.412i)T + (0.587 - 0.809i)T^{2} \) |
| 71 | \( 1 + (0.5 + 1.53i)T + (-0.809 + 0.587i)T^{2} \) |
| 73 | \( 1 + (0.951 - 0.309i)T^{2} \) |
| 79 | \( 1 + (0.809 - 0.587i)T^{2} \) |
| 83 | \( 1 + (-0.587 + 0.809i)T^{2} \) |
| 89 | \( 1 + (0.363 - 0.5i)T + (-0.309 - 0.951i)T^{2} \) |
| 97 | \( 1 + (0.896 - 1.76i)T + (-0.587 - 0.809i)T^{2} \) |
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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
−8.982991047575409140341699149058, −7.898341737628156263913217460608, −7.21750502634831771386215362765, −6.57360650017422792001627908528, −6.07538218975742500158783553660, −5.33104587774099527298733548018, −4.05131709510177680143412021467, −2.96877562441527023071446164379, −2.54589570566218207007203370501, −1.10339402427894535005551570400,
1.20782085047080440385464070045, 2.33494449684898013815295087693, 3.21517113233525612745362183440, 4.34062197131309767804191279687, 5.20505421829981140995199430561, 5.71822317352587418222198568359, 6.72988649097654986278823594801, 7.40548454048762069068996129022, 8.283425615242074139406316821820, 8.683648683371853369694088575914