| L(s) = 1 | + 5-s + 11-s − 2·13-s − 4·17-s − 2·23-s + 25-s + 8·31-s − 12·37-s − 2·41-s + 12·43-s + 12·53-s + 55-s + 6·59-s − 10·61-s − 2·65-s + 8·67-s + 8·71-s + 14·79-s + 14·83-s − 4·85-s + 4·89-s + 14·97-s + 101-s + 103-s + 107-s + 109-s + 113-s + ⋯ |
| L(s) = 1 | + 0.447·5-s + 0.301·11-s − 0.554·13-s − 0.970·17-s − 0.417·23-s + 1/5·25-s + 1.43·31-s − 1.97·37-s − 0.312·41-s + 1.82·43-s + 1.64·53-s + 0.134·55-s + 0.781·59-s − 1.28·61-s − 0.248·65-s + 0.977·67-s + 0.949·71-s + 1.57·79-s + 1.53·83-s − 0.433·85-s + 0.423·89-s + 1.42·97-s + 0.0995·101-s + 0.0985·103-s + 0.0966·107-s + 0.0957·109-s + 0.0940·113-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 388080 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 388080 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.072373845\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.072373845\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 - T \) |
| 7 | \( 1 \) |
| 11 | \( 1 - T \) |
| good | 13 | \( 1 + 2 T + p T^{2} \) |
| 17 | \( 1 + 4 T + p T^{2} \) |
| 19 | \( 1 + p T^{2} \) |
| 23 | \( 1 + 2 T + p T^{2} \) |
| 29 | \( 1 + p T^{2} \) |
| 31 | \( 1 - 8 T + p T^{2} \) |
| 37 | \( 1 + 12 T + p T^{2} \) |
| 41 | \( 1 + 2 T + p T^{2} \) |
| 43 | \( 1 - 12 T + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 - 12 T + p T^{2} \) |
| 59 | \( 1 - 6 T + p T^{2} \) |
| 61 | \( 1 + 10 T + p T^{2} \) |
| 67 | \( 1 - 8 T + p T^{2} \) |
| 71 | \( 1 - 8 T + p T^{2} \) |
| 73 | \( 1 + p T^{2} \) |
| 79 | \( 1 - 14 T + p T^{2} \) |
| 83 | \( 1 - 14 T + p T^{2} \) |
| 89 | \( 1 - 4 T + p T^{2} \) |
| 97 | \( 1 - 14 T + p T^{2} \) |
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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
−12.39841530775433, −12.02993106476596, −11.65812012358570, −11.07440947001113, −10.50461271457210, −10.34140381094279, −9.731123148730470, −9.296047526622959, −8.833874345719638, −8.488048681557213, −7.917513314413862, −7.301333042082249, −6.997993554625338, −6.363942109475312, −6.119213429461342, −5.468984583104039, −4.850397861160412, −4.670904594910858, −3.852230613781986, −3.537271151620142, −2.752313309356631, −2.124916681560086, −2.010188821470374, −0.9625357064729423, −0.5124025848350656,
0.5124025848350656, 0.9625357064729423, 2.010188821470374, 2.124916681560086, 2.752313309356631, 3.537271151620142, 3.852230613781986, 4.670904594910858, 4.850397861160412, 5.468984583104039, 6.119213429461342, 6.363942109475312, 6.997993554625338, 7.301333042082249, 7.917513314413862, 8.488048681557213, 8.833874345719638, 9.296047526622959, 9.731123148730470, 10.34140381094279, 10.50461271457210, 11.07440947001113, 11.65812012358570, 12.02993106476596, 12.39841530775433
