| L(s) = 1 | + 2-s − 3-s + 4-s − 6-s + 8-s + 9-s + 2·11-s − 12-s − 13-s + 16-s + 17-s + 18-s − 4·19-s + 2·22-s + 7·23-s − 24-s − 26-s − 27-s + 29-s − 3·31-s + 32-s − 2·33-s + 34-s + 36-s − 6·37-s − 4·38-s + 39-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 0.577·3-s + 1/2·4-s − 0.408·6-s + 0.353·8-s + 1/3·9-s + 0.603·11-s − 0.288·12-s − 0.277·13-s + 1/4·16-s + 0.242·17-s + 0.235·18-s − 0.917·19-s + 0.426·22-s + 1.45·23-s − 0.204·24-s − 0.196·26-s − 0.192·27-s + 0.185·29-s − 0.538·31-s + 0.176·32-s − 0.348·33-s + 0.171·34-s + 1/6·36-s − 0.986·37-s − 0.648·38-s + 0.160·39-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 7350 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7350 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.744591313\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.744591313\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + T \) |
| 5 | \( 1 \) |
| 7 | \( 1 \) |
| good | 11 | \( 1 - 2 T + p T^{2} \) |
| 13 | \( 1 + T + p T^{2} \) |
| 17 | \( 1 - T + p T^{2} \) |
| 19 | \( 1 + 4 T + p T^{2} \) |
| 23 | \( 1 - 7 T + p T^{2} \) |
| 29 | \( 1 - T + p T^{2} \) |
| 31 | \( 1 + 3 T + p T^{2} \) |
| 37 | \( 1 + 6 T + p T^{2} \) |
| 41 | \( 1 - 3 T + p T^{2} \) |
| 43 | \( 1 + T + p T^{2} \) |
| 47 | \( 1 - 12 T + p T^{2} \) |
| 53 | \( 1 - 11 T + p T^{2} \) |
| 59 | \( 1 - 3 T + p T^{2} \) |
| 61 | \( 1 + 5 T + p T^{2} \) |
| 67 | \( 1 + 12 T + p T^{2} \) |
| 71 | \( 1 - 4 T + p T^{2} \) |
| 73 | \( 1 - 14 T + p T^{2} \) |
| 79 | \( 1 + 2 T + p T^{2} \) |
| 83 | \( 1 - 3 T + p T^{2} \) |
| 89 | \( 1 + 10 T + p T^{2} \) |
| 97 | \( 1 - 10 T + p T^{2} \) |
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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
−7.60948694670136603940620247955, −7.02903446235978651686264467722, −6.48801993675004131322082246216, −5.70160156313535098776181831549, −5.13885314554339556196976403077, −4.36242280480095689501577341192, −3.73771642340924083873181453449, −2.79554234638343370401910461009, −1.85437527162664258587104211766, −0.78432917618571030579971049759,
0.78432917618571030579971049759, 1.85437527162664258587104211766, 2.79554234638343370401910461009, 3.73771642340924083873181453449, 4.36242280480095689501577341192, 5.13885314554339556196976403077, 5.70160156313535098776181831549, 6.48801993675004131322082246216, 7.02903446235978651686264467722, 7.60948694670136603940620247955
