| L(s) = 1 | + i·3-s + (2.15 + 0.594i)5-s − 0.633i·7-s − 9-s − 0.177·11-s − i·13-s + (−0.594 + 2.15i)15-s + 4.48i·17-s + 0.633·21-s + 6.48i·23-s + (4.29 + 2.56i)25-s − i·27-s + 3.49·29-s − 0.177i·33-s + (0.376 − 1.36i)35-s + ⋯ |
| L(s) = 1 | + 0.577i·3-s + (0.964 + 0.265i)5-s − 0.239i·7-s − 0.333·9-s − 0.0536·11-s − 0.277i·13-s + (−0.153 + 0.556i)15-s + 1.08i·17-s + 0.138·21-s + 1.35i·23-s + (0.858 + 0.512i)25-s − 0.192i·27-s + 0.649·29-s − 0.0309i·33-s + (0.0636 − 0.230i)35-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1560 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.265 - 0.964i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1560 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.265 - 0.964i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.939380779\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.939380779\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 - iT \) |
| 5 | \( 1 + (-2.15 - 0.594i)T \) |
| 13 | \( 1 + iT \) |
| good | 7 | \( 1 + 0.633iT - 7T^{2} \) |
| 11 | \( 1 + 0.177T + 11T^{2} \) |
| 17 | \( 1 - 4.48iT - 17T^{2} \) |
| 19 | \( 1 + 19T^{2} \) |
| 23 | \( 1 - 6.48iT - 23T^{2} \) |
| 29 | \( 1 - 3.49T + 29T^{2} \) |
| 31 | \( 1 + 31T^{2} \) |
| 37 | \( 1 - 1.75iT - 37T^{2} \) |
| 41 | \( 1 - 7.67T + 41T^{2} \) |
| 43 | \( 1 - 7.49iT - 43T^{2} \) |
| 47 | \( 1 + 3.18iT - 47T^{2} \) |
| 53 | \( 1 - 1.75iT - 53T^{2} \) |
| 59 | \( 1 + 3.93T + 59T^{2} \) |
| 61 | \( 1 - 3.01T + 61T^{2} \) |
| 67 | \( 1 + 1.62iT - 67T^{2} \) |
| 71 | \( 1 + 4.41T + 71T^{2} \) |
| 73 | \( 1 - 3.85iT - 73T^{2} \) |
| 79 | \( 1 + 9.63T + 79T^{2} \) |
| 83 | \( 1 - 1.72iT - 83T^{2} \) |
| 89 | \( 1 - 0.589T + 89T^{2} \) |
| 97 | \( 1 + 4.45iT - 97T^{2} \) |
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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
−9.692427311840901962786848361521, −8.977595750834322779508845025698, −8.088290922271714578778787818756, −7.16742386198199297737769347541, −6.14102467561908513657094846516, −5.61966278316443831940758976743, −4.61549318511311562969175021557, −3.61320470501628728395107048276, −2.63943284682568153504071305274, −1.39365694010627929299672744409,
0.809061225040323970689081230590, 2.13812488778736864777687877639, 2.83842997439589926606777093322, 4.37782968954208533076445685167, 5.25459817319717347355018461694, 6.06870684733504609194791227539, 6.78023078848749131610140321851, 7.59754695930764970202729453684, 8.679691292675743622681983998858, 9.101025415288334076923246789041
