| L(s) = 1 | + 5.24·2-s − 1.98·3-s + 19.5·4-s + 1.36·5-s − 10.4·6-s + 60.3·8-s − 23.0·9-s + 7.14·10-s − 18.4·11-s − 38.8·12-s + 21.2·13-s − 2.71·15-s + 160.·16-s + 126.·17-s − 120.·18-s − 50.2·19-s + 26.5·20-s − 96.8·22-s − 82.1·23-s − 120.·24-s − 123.·25-s + 111.·26-s + 99.5·27-s + 58.4·29-s − 14.2·30-s + 319.·31-s + 358.·32-s + ⋯ |
| L(s) = 1 | + 1.85·2-s − 0.382·3-s + 2.43·4-s + 0.121·5-s − 0.710·6-s + 2.66·8-s − 0.853·9-s + 0.225·10-s − 0.506·11-s − 0.933·12-s + 0.453·13-s − 0.0466·15-s + 2.50·16-s + 1.79·17-s − 1.58·18-s − 0.607·19-s + 0.297·20-s − 0.938·22-s − 0.745·23-s − 1.02·24-s − 0.985·25-s + 0.840·26-s + 0.709·27-s + 0.374·29-s − 0.0865·30-s + 1.85·31-s + 1.98·32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2401 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2401 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(7.148641464\) |
| \(L(\frac12)\) |
\(\approx\) |
\(7.148641464\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 7 | \( 1 \) |
| good | 2 | \( 1 - 5.24T + 8T^{2} \) |
| 3 | \( 1 + 1.98T + 27T^{2} \) |
| 5 | \( 1 - 1.36T + 125T^{2} \) |
| 11 | \( 1 + 18.4T + 1.33e3T^{2} \) |
| 13 | \( 1 - 21.2T + 2.19e3T^{2} \) |
| 17 | \( 1 - 126.T + 4.91e3T^{2} \) |
| 19 | \( 1 + 50.2T + 6.85e3T^{2} \) |
| 23 | \( 1 + 82.1T + 1.21e4T^{2} \) |
| 29 | \( 1 - 58.4T + 2.43e4T^{2} \) |
| 31 | \( 1 - 319.T + 2.97e4T^{2} \) |
| 37 | \( 1 - 266.T + 5.06e4T^{2} \) |
| 41 | \( 1 - 246.T + 6.89e4T^{2} \) |
| 43 | \( 1 - 360.T + 7.95e4T^{2} \) |
| 47 | \( 1 - 139.T + 1.03e5T^{2} \) |
| 53 | \( 1 + 306.T + 1.48e5T^{2} \) |
| 59 | \( 1 + 562.T + 2.05e5T^{2} \) |
| 61 | \( 1 - 623.T + 2.26e5T^{2} \) |
| 67 | \( 1 - 923.T + 3.00e5T^{2} \) |
| 71 | \( 1 - 196.T + 3.57e5T^{2} \) |
| 73 | \( 1 - 62.3T + 3.89e5T^{2} \) |
| 79 | \( 1 + 432.T + 4.93e5T^{2} \) |
| 83 | \( 1 - 747.T + 5.71e5T^{2} \) |
| 89 | \( 1 + 327.T + 7.04e5T^{2} \) |
| 97 | \( 1 + 1.53e3T + 9.12e5T^{2} \) |
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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
−8.153288907149041175047422080300, −7.77519034689702647007166850340, −6.56739571885457325068311863525, −5.91100072239172837277669926634, −5.61549776002997974466110348991, −4.66016566845291420913709333578, −3.88903363649570049173360860482, −2.97548341281786015061214680086, −2.30893981169068211179936386267, −0.916226861971033656968419419756,
0.916226861971033656968419419756, 2.30893981169068211179936386267, 2.97548341281786015061214680086, 3.88903363649570049173360860482, 4.66016566845291420913709333578, 5.61549776002997974466110348991, 5.91100072239172837277669926634, 6.56739571885457325068311863525, 7.77519034689702647007166850340, 8.153288907149041175047422080300
