L(s) = 1 | + (−1.30 − 1.30i)2-s + 2.41i·4-s + (1.30 − 1.30i)7-s + (1.84 − 1.84i)8-s − i·11-s + (−0.541 − 0.541i)13-s − 3.41·14-s − 2.41·16-s + (−0.541 − 0.541i)17-s + (−1.30 + 1.30i)22-s + 1.41i·26-s + (3.15 + 3.15i)28-s − 1.41·31-s + (1.30 + 1.30i)32-s + 1.41i·34-s + ⋯ |
L(s) = 1 | + (−1.30 − 1.30i)2-s + 2.41i·4-s + (1.30 − 1.30i)7-s + (1.84 − 1.84i)8-s − i·11-s + (−0.541 − 0.541i)13-s − 3.41·14-s − 2.41·16-s + (−0.541 − 0.541i)17-s + (−1.30 + 1.30i)22-s + 1.41i·26-s + (3.15 + 3.15i)28-s − 1.41·31-s + (1.30 + 1.30i)32-s + 1.41i·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2475 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.998 + 0.0618i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2475 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.998 + 0.0618i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.5822914078\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.5822914078\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 \) |
| 5 | \( 1 \) |
| 11 | \( 1 + iT \) |
good | 2 | \( 1 + (1.30 + 1.30i)T + iT^{2} \) |
| 7 | \( 1 + (-1.30 + 1.30i)T - iT^{2} \) |
| 13 | \( 1 + (0.541 + 0.541i)T + iT^{2} \) |
| 17 | \( 1 + (0.541 + 0.541i)T + iT^{2} \) |
| 19 | \( 1 + T^{2} \) |
| 23 | \( 1 + iT^{2} \) |
| 29 | \( 1 - T^{2} \) |
| 31 | \( 1 + 1.41T + T^{2} \) |
| 37 | \( 1 + iT^{2} \) |
| 41 | \( 1 + T^{2} \) |
| 43 | \( 1 + (-0.541 - 0.541i)T + iT^{2} \) |
| 47 | \( 1 - iT^{2} \) |
| 53 | \( 1 + iT^{2} \) |
| 59 | \( 1 - 1.41T + T^{2} \) |
| 61 | \( 1 - T^{2} \) |
| 67 | \( 1 + iT^{2} \) |
| 71 | \( 1 - 1.41iT - T^{2} \) |
| 73 | \( 1 + (1.30 + 1.30i)T + iT^{2} \) |
| 79 | \( 1 + T^{2} \) |
| 83 | \( 1 + (1.30 - 1.30i)T - iT^{2} \) |
| 89 | \( 1 - 2T + T^{2} \) |
| 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
−8.784332020585509498225187529768, −8.169575581766859839732288355760, −7.55667893662220146706900403869, −6.98011448857517248932539613256, −5.43011880654021234982095405399, −4.39027265073625677684542111894, −3.63903271889435471938037318270, −2.64821200985332310567864560363, −1.59365133576891881507942923276, −0.61806714762758565946178062062,
1.67437467459614384415566365133, 2.22718725748870408080762175142, 4.35972686522240371958007260964, 5.15208369382608252189800019867, 5.71544130435856815309047991724, 6.63651416397258945035687474534, 7.40228129106809915705114558915, 7.922571076410726337939132066561, 8.895187136388233813810158858958, 8.969415239948821280139248417414