| L(s) = 1 | + (−1.22 − 1.22i)7-s + (−1.22 + 1.22i)13-s − i·19-s + 4·31-s + (−8.57 − 8.57i)37-s + (−7.34 + 7.34i)43-s − 4i·49-s − 13·61-s + (−8.57 − 8.57i)67-s + (−1.22 + 1.22i)73-s + 13i·79-s + 2.99·91-s + (−13.4 − 13.4i)97-s + (−13.4 + 13.4i)103-s − 2i·109-s + ⋯ |
| L(s) = 1 | + (−0.462 − 0.462i)7-s + (−0.339 + 0.339i)13-s − 0.229i·19-s + 0.718·31-s + (−1.40 − 1.40i)37-s + (−1.12 + 1.12i)43-s − 0.571i·49-s − 1.66·61-s + (−1.04 − 1.04i)67-s + (−0.143 + 0.143i)73-s + 1.46i·79-s + 0.314·91-s + (−1.36 − 1.36i)97-s + (−1.32 + 1.32i)103-s − 0.191i·109-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2700 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.973 + 0.229i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2700 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.973 + 0.229i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.3075060145\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.3075060145\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 5 | \( 1 \) |
| good | 7 | \( 1 + (1.22 + 1.22i)T + 7iT^{2} \) |
| 11 | \( 1 - 11T^{2} \) |
| 13 | \( 1 + (1.22 - 1.22i)T - 13iT^{2} \) |
| 17 | \( 1 - 17iT^{2} \) |
| 19 | \( 1 + iT - 19T^{2} \) |
| 23 | \( 1 + 23iT^{2} \) |
| 29 | \( 1 + 29T^{2} \) |
| 31 | \( 1 - 4T + 31T^{2} \) |
| 37 | \( 1 + (8.57 + 8.57i)T + 37iT^{2} \) |
| 41 | \( 1 - 41T^{2} \) |
| 43 | \( 1 + (7.34 - 7.34i)T - 43iT^{2} \) |
| 47 | \( 1 - 47iT^{2} \) |
| 53 | \( 1 + 53iT^{2} \) |
| 59 | \( 1 + 59T^{2} \) |
| 61 | \( 1 + 13T + 61T^{2} \) |
| 67 | \( 1 + (8.57 + 8.57i)T + 67iT^{2} \) |
| 71 | \( 1 - 71T^{2} \) |
| 73 | \( 1 + (1.22 - 1.22i)T - 73iT^{2} \) |
| 79 | \( 1 - 13iT - 79T^{2} \) |
| 83 | \( 1 + 83iT^{2} \) |
| 89 | \( 1 + 89T^{2} \) |
| 97 | \( 1 + (13.4 + 13.4i)T + 97iT^{2} \) |
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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
−8.519443263539469724897301531133, −7.66230900757160139406359521205, −6.94270783229284760360068374608, −6.32178730247947803427117611398, −5.33686793606065672390337845389, −4.51020769979082631237528883408, −3.64452420725221794865231753714, −2.74123509149996179003826995039, −1.55802670330172780053549263703, −0.096900487829214161724247720775,
1.51383653024074752224996406247, 2.71223066775721335953264370213, 3.42903771823153191837452436696, 4.54806977648822520497012746904, 5.33313157076125391821835001885, 6.15010733811767196456612589066, 6.85744281350140932722157673797, 7.70008655875368717501002035173, 8.507633790612188587302805586267, 9.103971095980835835409664814292
