| L(s) = 1 | + 2-s + 3-s + 4-s + 4·5-s + 6-s + 4.44·7-s + 8-s + 9-s + 4·10-s + 4.89·11-s + 12-s − 6.89·13-s + 4.44·14-s + 4·15-s + 16-s − 4.89·17-s + 18-s − 4.44·19-s + 4·20-s + 4.44·21-s + 4.89·22-s − 23-s + 24-s + 11·25-s − 6.89·26-s + 27-s + 4.44·28-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 0.577·3-s + 0.5·4-s + 1.78·5-s + 0.408·6-s + 1.68·7-s + 0.353·8-s + 0.333·9-s + 1.26·10-s + 1.47·11-s + 0.288·12-s − 1.91·13-s + 1.18·14-s + 1.03·15-s + 0.250·16-s − 1.18·17-s + 0.235·18-s − 1.02·19-s + 0.894·20-s + 0.970·21-s + 1.04·22-s − 0.208·23-s + 0.204·24-s + 2.20·25-s − 1.35·26-s + 0.192·27-s + 0.840·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4002 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4002 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(6.211084389\) |
| \(L(\frac12)\) |
\(\approx\) |
\(6.211084389\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 - T \) |
| 23 | \( 1 + T \) |
| 29 | \( 1 - T \) |
| good | 5 | \( 1 - 4T + 5T^{2} \) |
| 7 | \( 1 - 4.44T + 7T^{2} \) |
| 11 | \( 1 - 4.89T + 11T^{2} \) |
| 13 | \( 1 + 6.89T + 13T^{2} \) |
| 17 | \( 1 + 4.89T + 17T^{2} \) |
| 19 | \( 1 + 4.44T + 19T^{2} \) |
| 31 | \( 1 + 6T + 31T^{2} \) |
| 37 | \( 1 + 2T + 37T^{2} \) |
| 41 | \( 1 + 8.89T + 41T^{2} \) |
| 43 | \( 1 - 12.4T + 43T^{2} \) |
| 47 | \( 1 - 4T + 47T^{2} \) |
| 53 | \( 1 + 8.89T + 53T^{2} \) |
| 59 | \( 1 + 4.89T + 59T^{2} \) |
| 61 | \( 1 - 2.89T + 61T^{2} \) |
| 67 | \( 1 - 2T + 67T^{2} \) |
| 71 | \( 1 + 10.8T + 71T^{2} \) |
| 73 | \( 1 + 6.89T + 73T^{2} \) |
| 79 | \( 1 - 10T + 79T^{2} \) |
| 83 | \( 1 + 1.55T + 83T^{2} \) |
| 89 | \( 1 - 12.8T + 89T^{2} \) |
| 97 | \( 1 + 19.1T + 97T^{2} \) |
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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
−8.632939848226437503780191135231, −7.55197030579555448635151370596, −6.88355179466220170772472571999, −6.20688874064931707119253342254, −5.30488739505490042481655884485, −4.69873326243756692345218101486, −4.12525737397461835889529624963, −2.63328223493268613801757118097, −2.00573873507345237659337128524, −1.57893789606784592363738379877,
1.57893789606784592363738379877, 2.00573873507345237659337128524, 2.63328223493268613801757118097, 4.12525737397461835889529624963, 4.69873326243756692345218101486, 5.30488739505490042481655884485, 6.20688874064931707119253342254, 6.88355179466220170772472571999, 7.55197030579555448635151370596, 8.632939848226437503780191135231
