| L(s) = 1 | − 2-s + 4-s + 7-s − 8-s − 4.89·11-s + 0.449·13-s − 14-s + 16-s + 2·17-s + 6.44·19-s + 4.89·22-s − 6.89·23-s − 0.449·26-s + 28-s + 2.89·29-s − 0.898·31-s − 32-s − 2·34-s − 2·37-s − 6.44·38-s + 10.8·41-s − 8.89·43-s − 4.89·44-s + 6.89·46-s − 0.898·47-s + 49-s + 0.449·52-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 0.5·4-s + 0.377·7-s − 0.353·8-s − 1.47·11-s + 0.124·13-s − 0.267·14-s + 0.250·16-s + 0.485·17-s + 1.47·19-s + 1.04·22-s − 1.43·23-s − 0.0881·26-s + 0.188·28-s + 0.538·29-s − 0.161·31-s − 0.176·32-s − 0.342·34-s − 0.328·37-s − 1.04·38-s + 1.70·41-s − 1.35·43-s − 0.738·44-s + 1.01·46-s − 0.131·47-s + 0.142·49-s + 0.0623·52-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3150 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3150 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.191303430\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.191303430\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + T \) |
| 3 | \( 1 \) |
| 5 | \( 1 \) |
| 7 | \( 1 - T \) |
| good | 11 | \( 1 + 4.89T + 11T^{2} \) |
| 13 | \( 1 - 0.449T + 13T^{2} \) |
| 17 | \( 1 - 2T + 17T^{2} \) |
| 19 | \( 1 - 6.44T + 19T^{2} \) |
| 23 | \( 1 + 6.89T + 23T^{2} \) |
| 29 | \( 1 - 2.89T + 29T^{2} \) |
| 31 | \( 1 + 0.898T + 31T^{2} \) |
| 37 | \( 1 + 2T + 37T^{2} \) |
| 41 | \( 1 - 10.8T + 41T^{2} \) |
| 43 | \( 1 + 8.89T + 43T^{2} \) |
| 47 | \( 1 + 0.898T + 47T^{2} \) |
| 53 | \( 1 + 1.10T + 53T^{2} \) |
| 59 | \( 1 - 6.44T + 59T^{2} \) |
| 61 | \( 1 - 8.44T + 61T^{2} \) |
| 67 | \( 1 - 8T + 67T^{2} \) |
| 71 | \( 1 - 10.8T + 71T^{2} \) |
| 73 | \( 1 - 6.89T + 73T^{2} \) |
| 79 | \( 1 + 2.89T + 79T^{2} \) |
| 83 | \( 1 - 2.44T + 83T^{2} \) |
| 89 | \( 1 - 10T + 89T^{2} \) |
| 97 | \( 1 - 3.79T + 97T^{2} \) |
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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
−8.482695314716223891060835874297, −7.954365024723372766470644006249, −7.50278031899873678476055597616, −6.55043219412069728060396235264, −5.55099238941352453213462199316, −5.10518644789293325280282707379, −3.83692107553771915336431592214, −2.86580594149057280301441010398, −1.98663644315735309451399381304, −0.72590807720796895153372359854,
0.72590807720796895153372359854, 1.98663644315735309451399381304, 2.86580594149057280301441010398, 3.83692107553771915336431592214, 5.10518644789293325280282707379, 5.55099238941352453213462199316, 6.55043219412069728060396235264, 7.50278031899873678476055597616, 7.954365024723372766470644006249, 8.482695314716223891060835874297
