L(s) = 1 | + (−0.366 + 0.366i)2-s + 0.732i·4-s + (0.707 + 0.707i)7-s + (−0.633 − 0.633i)8-s + 1.93i·11-s − 0.517·14-s − 0.267·16-s + (−0.707 − 0.707i)22-s + (−1.36 − 1.36i)23-s + (−0.517 + 0.517i)28-s + 0.517·29-s + (0.732 − 0.732i)32-s + (0.707 + 0.707i)37-s + (−0.707 + 0.707i)43-s − 1.41·44-s + ⋯ |
L(s) = 1 | + (−0.366 + 0.366i)2-s + 0.732i·4-s + (0.707 + 0.707i)7-s + (−0.633 − 0.633i)8-s + 1.93i·11-s − 0.517·14-s − 0.267·16-s + (−0.707 − 0.707i)22-s + (−1.36 − 1.36i)23-s + (−0.517 + 0.517i)28-s + 0.517·29-s + (0.732 − 0.732i)32-s + (0.707 + 0.707i)37-s + (−0.707 + 0.707i)43-s − 1.41·44-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1575 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.583 - 0.812i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1575 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.583 - 0.812i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.8699717508\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.8699717508\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 \) |
| 5 | \( 1 \) |
| 7 | \( 1 + (-0.707 - 0.707i)T \) |
good | 2 | \( 1 + (0.366 - 0.366i)T - iT^{2} \) |
| 11 | \( 1 - 1.93iT - T^{2} \) |
| 13 | \( 1 + iT^{2} \) |
| 17 | \( 1 + iT^{2} \) |
| 19 | \( 1 + T^{2} \) |
| 23 | \( 1 + (1.36 + 1.36i)T + iT^{2} \) |
| 29 | \( 1 - 0.517T + T^{2} \) |
| 31 | \( 1 - T^{2} \) |
| 37 | \( 1 + (-0.707 - 0.707i)T + iT^{2} \) |
| 41 | \( 1 + T^{2} \) |
| 43 | \( 1 + (0.707 - 0.707i)T - iT^{2} \) |
| 47 | \( 1 + iT^{2} \) |
| 53 | \( 1 + (-1 - i)T + iT^{2} \) |
| 59 | \( 1 - T^{2} \) |
| 61 | \( 1 - T^{2} \) |
| 67 | \( 1 + (1.22 + 1.22i)T + iT^{2} \) |
| 71 | \( 1 - 0.517iT - T^{2} \) |
| 73 | \( 1 + iT^{2} \) |
| 79 | \( 1 + iT - T^{2} \) |
| 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
−9.751157767346415551629491751600, −8.978960158951648640334258790602, −8.193192952355617256424148355955, −7.67253603800835825057058636670, −6.82171107798779923581676283463, −6.04223698347803549483599558539, −4.71719847037648989923969670203, −4.27341831652717955657036709063, −2.79711425493672261724523300334, −1.92369650854773894545859383406,
0.790563722286526069708630232352, 1.91367649840646536981872745317, 3.24237934294312350315762593549, 4.21554889279392513614667198906, 5.46309950015762734860962019419, 5.85972655628513521156961662736, 6.93469527039154666004994314650, 8.042835155004543134045134854225, 8.517030376886790596985894533800, 9.429945262509394613711219070863