| L(s) = 1 | − 2-s + 4-s + 1.58·5-s − 8-s − 1.58·10-s − 11-s − 4.24·13-s + 16-s + 2.82·17-s − 0.171·19-s + 1.58·20-s + 22-s − 3.24·23-s − 2.48·25-s + 4.24·26-s + 8.24·29-s + 1.24·31-s − 32-s − 2.82·34-s − 3.24·37-s + 0.171·38-s − 1.58·40-s + 11.8·41-s + 10.4·43-s − 44-s + 3.24·46-s + 0.343·47-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 0.5·4-s + 0.709·5-s − 0.353·8-s − 0.501·10-s − 0.301·11-s − 1.17·13-s + 0.250·16-s + 0.685·17-s − 0.0393·19-s + 0.354·20-s + 0.213·22-s − 0.676·23-s − 0.497·25-s + 0.832·26-s + 1.53·29-s + 0.223·31-s − 0.176·32-s − 0.485·34-s − 0.533·37-s + 0.0278·38-s − 0.250·40-s + 1.84·41-s + 1.59·43-s − 0.150·44-s + 0.478·46-s + 0.0500·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2646 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2646 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.353306323\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.353306323\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) |
| 7 | \( 1 \) |
| good | 5 | \( 1 - 1.58T + 5T^{2} \) |
| 11 | \( 1 + T + 11T^{2} \) |
| 13 | \( 1 + 4.24T + 13T^{2} \) |
| 17 | \( 1 - 2.82T + 17T^{2} \) |
| 19 | \( 1 + 0.171T + 19T^{2} \) |
| 23 | \( 1 + 3.24T + 23T^{2} \) |
| 29 | \( 1 - 8.24T + 29T^{2} \) |
| 31 | \( 1 - 1.24T + 31T^{2} \) |
| 37 | \( 1 + 3.24T + 37T^{2} \) |
| 41 | \( 1 - 11.8T + 41T^{2} \) |
| 43 | \( 1 - 10.4T + 43T^{2} \) |
| 47 | \( 1 - 0.343T + 47T^{2} \) |
| 53 | \( 1 - 8T + 53T^{2} \) |
| 59 | \( 1 - 0.343T + 59T^{2} \) |
| 61 | \( 1 - 9.17T + 61T^{2} \) |
| 67 | \( 1 + 10.2T + 67T^{2} \) |
| 71 | \( 1 - 7.24T + 71T^{2} \) |
| 73 | \( 1 + 8.82T + 73T^{2} \) |
| 79 | \( 1 - 10.4T + 79T^{2} \) |
| 83 | \( 1 - 15.8T + 83T^{2} \) |
| 89 | \( 1 - 14.6T + 89T^{2} \) |
| 97 | \( 1 - 11.3T + 97T^{2} \) |
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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
−9.048121711292828216769147804242, −7.978928256670565543487052484266, −7.57215610758133581238845957492, −6.61263080496247575370758533277, −5.86209792688887011328768440438, −5.12699870463752234863763696541, −4.06301884384114683166555557794, −2.74848811390630812162574798469, −2.14357713046529586958414535077, −0.803817008295307276716934406956,
0.803817008295307276716934406956, 2.14357713046529586958414535077, 2.74848811390630812162574798469, 4.06301884384114683166555557794, 5.12699870463752234863763696541, 5.86209792688887011328768440438, 6.61263080496247575370758533277, 7.57215610758133581238845957492, 7.978928256670565543487052484266, 9.048121711292828216769147804242
