| L(s) = 1 | + 2-s + 2.01·3-s + 4-s + 5-s + 2.01·6-s + 1.08·7-s + 8-s + 1.06·9-s + 10-s + 1.33·11-s + 2.01·12-s + 3.13·13-s + 1.08·14-s + 2.01·15-s + 16-s − 1.91·17-s + 1.06·18-s − 3.62·19-s + 20-s + 2.19·21-s + 1.33·22-s + 3.32·23-s + 2.01·24-s + 25-s + 3.13·26-s − 3.89·27-s + 1.08·28-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 1.16·3-s + 0.5·4-s + 0.447·5-s + 0.823·6-s + 0.411·7-s + 0.353·8-s + 0.355·9-s + 0.316·10-s + 0.403·11-s + 0.582·12-s + 0.869·13-s + 0.290·14-s + 0.520·15-s + 0.250·16-s − 0.465·17-s + 0.251·18-s − 0.831·19-s + 0.223·20-s + 0.478·21-s + 0.285·22-s + 0.694·23-s + 0.411·24-s + 0.200·25-s + 0.615·26-s − 0.750·27-s + 0.205·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6010 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6010 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(6.010986044\) |
| \(L(\frac12)\) |
\(\approx\) |
\(6.010986044\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - T \) |
| 5 | \( 1 - T \) |
| 601 | \( 1 - T \) |
| good | 3 | \( 1 - 2.01T + 3T^{2} \) |
| 7 | \( 1 - 1.08T + 7T^{2} \) |
| 11 | \( 1 - 1.33T + 11T^{2} \) |
| 13 | \( 1 - 3.13T + 13T^{2} \) |
| 17 | \( 1 + 1.91T + 17T^{2} \) |
| 19 | \( 1 + 3.62T + 19T^{2} \) |
| 23 | \( 1 - 3.32T + 23T^{2} \) |
| 29 | \( 1 - 1.71T + 29T^{2} \) |
| 31 | \( 1 - 2.69T + 31T^{2} \) |
| 37 | \( 1 - 10.2T + 37T^{2} \) |
| 41 | \( 1 + 6.17T + 41T^{2} \) |
| 43 | \( 1 - 8.74T + 43T^{2} \) |
| 47 | \( 1 - 2.84T + 47T^{2} \) |
| 53 | \( 1 + 4.76T + 53T^{2} \) |
| 59 | \( 1 - 13.6T + 59T^{2} \) |
| 61 | \( 1 + 12.2T + 61T^{2} \) |
| 67 | \( 1 - 12.1T + 67T^{2} \) |
| 71 | \( 1 + 7.00T + 71T^{2} \) |
| 73 | \( 1 - 4.86T + 73T^{2} \) |
| 79 | \( 1 + 3.44T + 79T^{2} \) |
| 83 | \( 1 - 3.36T + 83T^{2} \) |
| 89 | \( 1 - 13.7T + 89T^{2} \) |
| 97 | \( 1 + 4.64T + 97T^{2} \) |
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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
−8.167827276442703927248157021362, −7.42398257649942706757701760910, −6.49083712383683321694960774495, −6.07305636157585551120808194551, −5.07247923288090535439925754753, −4.29613153321790196424452373558, −3.64894991243308900983895566043, −2.77086172210161701279745226881, −2.16531965340644094655566678711, −1.19455713098122704185629408224,
1.19455713098122704185629408224, 2.16531965340644094655566678711, 2.77086172210161701279745226881, 3.64894991243308900983895566043, 4.29613153321790196424452373558, 5.07247923288090535439925754753, 6.07305636157585551120808194551, 6.49083712383683321694960774495, 7.42398257649942706757701760910, 8.167827276442703927248157021362
